What is... a grossone? Y. Sergeyev developed a positional system for representing infinite numbers using a basic unit called a "grossone", as well as what he calls an "infinity computer".  The mathematical value of this seems dubious but numerous articles have already appeared in refereed research journals.  Thus, there are currently 23 such articles in mathscinet not to speak of numerous lectures in conferences.
In a comment accessible here Sergeyev asserts that "Levi-Civita numbers are built using a generic infinitesimal $\varepsilon$ ... whereas our numerical computations with finite quantities are concrete and not generic."  Here apparently "finite" is a misprint and should be "infinite". How is this comment on the difference between Sergeyev's grossone one the one hand, and the Levi-Civita unit on the other, to be understood?
In a 2013 article, Sergeyev compares his grossone to Levi-Civita in the following terms in footnote 5: 5 At the first glance the numerals (7) can remind numbers from the Levi-Civita field (see [20]) that is a very interesting and important precedent of algebraic manipulations with infinities and infinitesimals. However, the two mathematical objects have several crucial differences. They have been introduced for different purposes by using two mathematical languages having different accuracies and on the basis of different methodological foundations. In fact, Levi-Civita does not discuss the distinction between numbers and numerals. His numbers have neither cardinal nor ordinal properties; they are build using a generic infinitesimal and only its rational powers are allowed; he uses symbol 1 in his construction; there is no any numeral system that would allow one to assign numerical values to these numbers; it is not explained how it would be possible to pass from d a generic infinitesimal h to a concrete one (see also the discussion above on the distinction between numbers and numerals). In no way the said above should be considered as a criticism with respect to results of Levi-Civita. The above discussion has been introduced in this text just to underline that we are in front of two different mathematical tools that should be used in different mathematical contexts. It would be interesting to have a specialist in numerical analysis comment on Sergeyev's use of the term "numerical" to explain the difference between his grossone and an infinite element of the Levi-Civita field.
Sergeyev claims that his grossone has the properties of both ordinal and cardinal numbers. Does he give a definition that would ensure such properties, or is this claim merely a declarative pronouncement?
Following the publication of an article by Sergeyev in EMS Surveys in Mathematical Sciences, the editors published the following clarification:

Statement of the editorial board
We deeply regret that this article appears in this issue of the EMS Surveys in Mathematical Sciences.
It was a serious mistake to accept it for publication. Owing to an unfortunate error, the entire processing of the paper, including the decision to accept it, took place without the editorial board being aware of what was happening. The editorial board unanimously dissociates itself from this decision. It is not representative of the very high level that we expect to see in our journal, which can be assessed from all other papers that we have published.
Both editors-in-chief have assumed responsibility for these mistakes and resigned from their position. Having said that, we add that this journal would not exist without their dedication and years of hard work, and we wish to register our thanks to them.

An interesting viewpoint of a computer scientist is developed here (as well as a related discussion of legal issues in the comments).
The unanimous statement of the EMS Surveys editors is now fleshed out in the Zentralblatt review and the MathSciNet review also available here.
Question.  Does Sergeyev's grossone admit a consistent interpretation?  Lolli in his answer argues that it does; others have argued otherwise.
 A: Mathematics  is the search of truth by way of proof, as defined by Mac Lane. Mathematics is not a collection of opinions or hyphotheses about what someone has meant.  
Sergeyev defines  his grossone as follows: "We have introduced grossone as the quantity of natural numbers. Thus, it is the biggest natural number... " See, for instance, Annales UMCS Informatica AI 4 (2006) p.26 
This is not what we encounter by the way of proof.
A: It is with some trepidation that I write in defense of Grossone when people who know much more  have spoken against it.
First, I would like to point out that Lolli's paper and Kauffman's paper both deal with a relaxation of Sergeyev's approach. Thus, even if Lolli's paper is indeed a reformulation of nonstandard arithmetic (and his answer indicates that he does not believe this to be the case), this has no bearing on the originality of Sergeyev's Grossone.  Kutateladze's review also discusses a nonstandard arithmetic object, which I don't think is the same as Grossone- for example, there is no bijection between a set with Grossone elements and a set with one fewer element, whereas I think there would be such a bijection with Kutateladze's object.
Sergeyev himself says that the main point about Grossone is that it is "numeric" and not "symbolic". I'm not entirely clear what he means concretely, but below is my thought.
As I see things, the important thing to realize about Grossone is that it is a notion suited to computer science and to applied mathematics. If one wants to axiomatize it then I suppose one could, but that's not Sergeyev's goal. Inside computer science for example, Grossone allows various Turing machine formalisms to be distinguished in a way that computer scientists have found useful, and that's what matters to Sergeyev. Can the same distinctions be made using Kauffman's "generic finite set"? Almost certainly yes- but that doesn't take away from Sergeyev's approach.  
One property of Grossone is that it is divisible by arbitrary natural numbers. Let's put on our "applied math hat" and not our "logic hat" and see why Grossone ought to have such a property. An "applied" infinity is a very large number. For a fixed natural number $m$ (say 2 or 3) and a "generic" natural number $N$, the noninteger part of $N/m$ is very small compared to its integer part. So an applied mathematician can safely ignore it (which is not to say that Grossone lacks a notion of infinitesimals). All properties of Grossone should be viewed in this light. Perhaps Grossone is just a "naïve infinite number" for applications, in the sense that it requires very little background to use correctly and it can easily be applied. 
In conclusion, note that Sergeyev's papers are published in computer science journals and in applied mathematics journals, but never in logic journals. Moreover, Grossone has indeed been well received. See for example this review of Andrew Adamatzky and other reviews listed on Sergeyev's site, all written by applied mathematicians and computer scientists. 
A: I do not understand what the bounty on this question is for, as it seems to me that the other answers were already rather devastating. Here is a semi-reasoned technical answer.
According to G. Lolli (the paper you cite) "Sergeyev is wary of the axiomatic method because he thinks that by adopting it we would be tied to the expressive power of a language in the description of mathematical objects and concepts." Serious mathematics requires serious adherence to the generally accepted standards of mathematics. Perhaps prof. Sergeyev thinks that he can surpass the limitations of formalization by taking a non-standard route to mathematics, but I would rather suspect that route will take him backwards in time and much closer to (a bad kind of) philosophy than most mathematicians would feel comfortable with.
Regarding the formalization by G. Lolli, I see no difference between what is done in the paper and non-standard arithmetic. A grossone $G$ is axiomatized by the infinitely many axioms $0 < G$, $1 < G$, $2 < G$, ... which is exactly how one can get non-standard arithmetic going. The paper does not even mention non-standard arithmetic. This is what you get for publishing logic papers in applied math journals.
So, it looks to me that grossones are a moving target with unclear and confused mathematical content, until one actually pins them down with a precise mathematical definition, only to find out they are not new at all.
Update: it was pointed out that none of the answers has commented on the computational part of the grossone theory. I had a look at three papers, found on the infinity computer web site:


*

*The recommended paper to start with is Sergeyev Ya.D. (2010) Lagrange Lecture: Methodology of numerical computations with infinities and infinitesimals, Rendiconti del Seminario Matematico dell'Università e del Politecnico di Torino, 68(2), 95–113. It has a lot of informal descriptions and philosophy, some illustrative examples, but nothing that would actually describe a revolutionary new way of computing. Rather, it looks like ideas that could possibly lead to re-invention of non-standard arithmetic.

*Sergeyev Ya.D. (2016) The exact (up to infinitesimals) infinite perimeter of the Koch snowflake and its finite area, Communications in Nonlinear Science and Numerical Simulation, 31(1–3):21–29. I tried this paper because the title promised that there would be a concrete result in it. There is, of course, but again the theory of computation underlying the method is not properly explained. There are examples and analogies which again hint at something like non-standard arithmetic.

*Sergeyev Ya.D. (2015) Computations with grossone-based infinities, C.S. Calude, M.J. Dinneen (Eds.), Proc. of the 14th International Conference “Unconventional Computation and Natural Computation”, Lecture Notes in Computer Science, vol. 9252, Springer,  89-106. A pattern starts to emerge. Every paper contains a very long introduction to the philosophy and ideas about grossones, supported by illustrative examples, but there is no clear explanation of what is going on.
All three papers present an equational system for grossones, i.e., things like associativity, commutativity, and other equations one would expect. A smart person can use these to simplify expressions and thereby "compute" results. But a computational model requires a description of a general procedure for performing computations, whatever it is. Is there a method for normalizing expressions involving grossones? Or perhaps an abstract machine one can run? Or something else?
I suppose the infinity computer is hiding in the patent. We shall never know. And I have now wasted more time on this than 50 points of bounty are worth. If someone can point me at an actual description of a computational model (whether it be "axiomatic" or not) which is not composed of a series of analogies and good ideas, I might take another look.
A: My original question focused on Professor Lolli's apparent advocacy of Sergeyev's work. Thanks to the efforts of several editors the picture with Lolli's article has become clearer to me, and I would like to summarize it in 8 points below below.
(1) Professor Lolli originally claimed that earlier work on nonstandard models involves nonconservative extensions of PA, whereas Professor Lolli's formalisation of the grossone was conservative.
(2) Editor Bauer provided a pointer to papers by Henson et al. (citing earlier work by Kreisel as well as H. Friedman), analyzing nonstandard conservative extensions of PA.
(3) Professor Lolli responded that those conservative extensions are too powerful for computer science.
(4) The modified claim referred to in (3) is certainly different from Professor Lolli's original claim that earlier work dealt with NONconservative extensions.
(5) Professor Lolli's comment does not deal with the question as to what might be the novelty of developing a weaker conservative extension than those already available in the literature.
(6) Professor Lolli does not address the question why, being apparently Abraham Robinson's student, he apparently overlooked explicit and multiple references in Robinson's book to earlier work by Thoralf Skolem on nonstandard models or arithmetic.
(7) As a disclaimer, it may be helpful if Professor Lolli could issue a clarification regarding the following issue.  A Russian translation of Lolli's book supervised by Sergeyev was released to great fanfare in Nizhnii Novgorod in the presence of the Italian ambassador, see http://www.unn.ru/news/?id=2036 
(8) However it seems to be impossible to get a copy of the Russian translation, which seems to have mysteriously disappeared, while Sergeyev claims on his webpage that it received a prize as a best translation of the year, see http://wwwinfo.dimes.unical.it/~yaro/list_pub.html (the comment there says: "Sergeyev Ya.D. (2012) Foreword of the scientific editor of the Russian translation of the book G. Lolli, Filosofia della matematica: L'eredita del Novecento, Series 'Scientific Classics', University of Nizhni Novgorod Press, Nizhni Novgorod. The winner in the annual national competition 'University Book' in Russia in the nomination 'The best translation'.")
A: In my paper about the Grossone, I point out that the logic of this formalism is identical (in my version) to using $1 + x + x^2 + ... + x^G$ as a FINITE SUM with 
$G$ a generic positive integer. One can then manipulate the series and
look at the limiting behaviour in many cases. There is no need to invoke any new concepts about infinity. This point of view may be at variance with the interpretations of Yaroslav for his invention, but I suggest that this is what is happening here. 
A: What I did in my paper on Sergeyev’s Grossone that has been mentioned in your discussion  was to present an axiomatised theory of arithmetic in the  language of Peano arithmetic augmented with a new constant for Grossone. I did it  because many colleagues seemed  to  think that  Sergeyev’s approach didn’t respect the standards of acceptable mathematical exposition.  In my theory I showed that  Sergeyev’s axioms are provable while I argued that  it  respected Sergeyev’s outlook expressed in his so called (by him) postulates. The main result is that the theory  is a conservative extension of Peano arithmetic, that is  that it proves the same  sentences in  the language without Grossone; hence Sergeyev theory, if my theory is faithful to his spirit, in consistent if Peano’s arithmetic is consistent.
Moreover this should show that Sergeyev’s methods, at least as to their arithmetical part,  are something different form non standard analysis.  To discuss non standard analysis is not easy, since one should be precise on what one means;  there are various approaches, among  whom Nelson’s Internal Set Theory IST is probably the smoothest.  But any theory for non standard methods should be stronger  than Sergeyev’s, since these methods do not  seem to be a conservative extension of the classical  ones;  one needs either third order logic or strong assumptions on the existence of  special ultrafilters. In Sergeyev’s theory there is no transfer principle, and so on.
There is another paper  you might be interested in:
 
F. Montagna, G. Simi and A. Sorbi, Taking the Pirahã seriously, Communication in Nonlinear Science and Numerical Simulation, 21(1–3), April 2015, 52–69.

Here, the authors  go deeper into the logical use of Grossone in relation to predicative arithmetic.
Andrej Bauer asked in the comments:
"Could you comment on how your result is related to results by Kreisel
on non-standard arithmetic being conservative over PA? See for example
Proposition 2.3 in jstor.org/stable/2274260 and the reference to
Kreisel therein? (I cannot find a direct link to Kreisel, sorry.)"
Our results are rather similar, also in the proof, by compactness and adjustment of models. But while I think that my theory can be considered a fair rendition of Sergeyev’s  outlook, it is  doubtful that Kreisel’s *PA can be considered a faithful framework for non standard arithmetic. The authors  of the paper  mentioned in the comment, Henson, Kaufmann and Keisler  after  briefly recalling Kreisel’s result (indeed the first in this line)  go on saying that most of non standard  arithmetical  results depend on the use of omega1-saturated models, and discuss possible strengthening  of the theory to approximate the properties of such models. 
Non standard mathematics is more ambitious than Sergeyev’s computational interest, and requires in consequence stronger logical principles. It seems to me that at present there is no consensus on a formalisation of these principles, but perhaps for Nelson’s set theory. 
Gabriele Lolli
A: A quick web search finds a fair amount of citations and analysis, e.g. http://arxiv.org/pdf/1401.7545v2.pdf
I didn't read much of it but it reminds me of surreal numbers, both in the sense of being a computing system with infinite and infinitesimal quantities, and being a mathematically legitimate and interesting concept but one that hasn't had really far-reaching consequences so far AFAIK.
Added: here's a not-so-favorable review:


*

*https://www.researchgate.net/profile/Semen_Kutateladze/publication/226817008_A_pennorth_of_grossone/links/00b7d51b130fb73111000000.pdf
A: I would like to summarize some findings concerning the mathematics of Sergeyev's grossone.
(1) Sergeyev's writing seems to contain confusion between the notions of ordinal
and a cardinal numbers.  These are the same in the finite case, but he
postulates that his grossone has properties of both ordinals and
cardinals.  This certainly sounds interesting.  However, he does not
provide any justification for such things. 
(2) Sergeyev's postulation rests at the level of a declarative
pronouncement rather than a mathematical development.  Perhaps his
intention is to develop a new mathematical foundation, but if so he
provides very little explanation.  Some of his comments here look a lot like freshman calculus errors.
(3) A number of experts have emphasized at this MO discussion that
whatever seems to be novel in Sergeyev's claims is subsumed under
nonstandard models of arithmetic.  Therefore there is nothing new here
except for a refusal to provide any details, as also noted by several
participants in the discussion.  It is fine to popularize ideas
related to infinity and infinitesimals.  Where is becomes problematic
is when other people's work is presented as his own insight.
(4) Sergeyev has repeatedly implied that he has superceded Cantor's
set theory, and makes various claims based on Greek mathematics to the
effect that the whole is greater than the part.  This is also mere
posturing on his part, and remains at the level of declarative
pronouncement that may appeal to the masses but does not impress
mathematicians who expect explanation rather than declaration.
(5) Specifically with regard to numerosities, note that all of
Sergeyev's work on infinity is posterior to that of di Nasso et al.
(2003).  Sergeyev claims his theory is different but it is only
different in that, unlike di Nasso, he does not justify any of his
claims.
(6) Similar remarks apply to Sergeyev's repeated comparisons of his
own "framework" with Robinson's, with the latter invariably coming out
looking inferior.
(7) Sometimes Sergeyev claims that his grossone is part of $\mathbb{N}$; on other occasions he claims that his grossone is the number of elements in $\mathbb N$.  If the circularity of this may bother some people that does not seem to include Sergeyev himself.
(8) Several reviewers for mathscinet have noted that the flood of
publications coming from Sergeyev is repetitous and of dubious
quality.  See for example a recent review by Leon Harkleroad at
http://www.ams.org/mathscinet-getitem?mr=3328558
(9) Sergeyev makes a distinction between "number" and "numeral".
There is a lot of verbiage in his writing about this distinction and a
lot seems to hinge on it.  Now it is true that a real number per se is
not implementable on the computer, and one needs to work with one or
another representation, be it decimal numerals or some other form.
That much is true and is obvious.  I was not able to discern any
additional meaning beyond the above in the flou artistique skillfully
made to envelope this distinction in Sergeyev's writing.
(10) Sergeyev traces the history of human engagement with number
systems from Pirahã to Cantor to Sergeyev in broad strokes: 
"With respect to our methodology, the mathematical results obtained by
Pirahã, Cantor, and those presented in this paper do not contradict to
each other." 
(see page 594 in http://www.mii.lt/informatica/pdf/INFO725.pdf) One wonders about the effectiveness of exploiting the Pirahã to "motivate" the grossone or any other form of infinity.
The above analysis of Sergeyev's "numerical infinity" was presented in more detailed form in this 2017 publication in Foundations of Science.
Sergeyev has since posted this feb 2018 rebuttal on the arXiv.
A: For those who don't believe in the Yaro computer I add a screenshot from 
http://www.spacephys.ru/kompyuter-beskonechnykh-vychislenii 
Does anybody see a suslik here?

