method of finding roots of polynominal equations with arithmetic operations and roots and other functions Lets recall Platonic construction in plane geometry. It is impossible to square a circle using only ruler and callipers. But is also known that it is possible to do it with ruler which has a mark on it.
So we know by Gaois theory that we can't write expressions for the solutions to all algebraic equations using the four arithmetic operations and nth roots. 
Question: is it possible to write expressions for the solutions to all algebraic equations using the four arithmetic operations and nth roots and something else in finite numbers of factors? For example adding logarithm to the set of operations? ( or other function, or class of functions...)? 
If Yes: Do they form any kind of algebraic structure?
 A: According to Kolmogorov's 1957 solution of Hilbert's 13th problem, "generalized slide-rules suffice" ... For given $n$, there exists a finite list of functions of one variable so that solutions for polynomial equations of degree $n$ can be written in terms of addition together with these functions.  Indeed, any continuous function of $n$ variables can be written this way (no need to mention polynomials). 
addition (statement taken from my book Classics on Fractals, p. 335)
There exist fixed continuous functions $\phi_{pq}(x)$ on $I = [0,1]$ so that each continuous function $f$ on $I^n$ can be written in the form
$$
f(x_1,\dots,x_n) = \sum_{q=1}^{2n+1}g_q\left(\sum_{p=1}^n \phi_{pq}(x_p)\right)
$$
where $g_q$ are properly chosen continuous functions of one variable.
A: Eisenstein in 1844 found a very simple hypergeometric series solution of the
quintic $x^5+x=a$, namely
$x=a-a^5+10\frac{a^9}{2!}-15\cdot 14 \frac{a^{13}}{3!}+20\cdot 19\cdot 18\frac{a^{17}}{4!}-\cdots$
Since it was shown by Bring in 1786 that the general quintic is reducible by
radicals to this form, Eisenstein's solution is probably the simplest solution
possible for the quintic.
For more on this story, see S.J. Patterson, Eisenstein and the quintic equation,
in Historia Mathematica 17 (1990), pp.132--140.
A: Well, the logarithm can be defined as an integral or locally as an infinite series. So if we are allowed to add functions defined in this matter to our class of functions, then the answer to your problem is trivially true. But of course, we need to add new functions for each degree.
Classically, the quintic equation (degree 5) was solved in this manner using jacobi theta functions, which can be defined using infinite products or Fourier series. In 1923 Birkeland derived hypergeometric series expansions of the roots. Later, in 1983 in an appendix to Mumfords 'Tata lectures on Theta' H. Umemura showed that the general algebraic equation can be solved using Riemann theta functions. This compares to solving the equation 
$$x^n-a$$ using $$x=\exp\big(\frac1{n} \ln a\big),$$ but with $\ln a$ replaced by an hyperelliptic integral and  $\exp$ replaced by a quotient of thetafunctions. For Umemura's article this here.
A: The answer is no: you need to keep adding more and more operations.  For degree $n=5$ you can use elliptic functions (or the Jacobi $\Theta$ function, or Bring radicals - see below), for $n=6,7$ the Lauricella functions are needed (they are a 2-variable version of hypergeometric functions), and after that you need more and more complicated 'new' functions.  You can apparently use the elliptic Siegel functions (aka Siegel modular forms) for the general case, but I've never looked into that.  Most of this was all worked out in gruesome detail by analysts for decades, with lots of research on this up until the early 20th century (and then generally forgotten!).
For trinomial equations, you can use hypergeometrics to solve all of them.  The derivation of that is fun.
It is hard to find information about this on the web.  The 'best' discussion of related ideas is the Wikipedia page on the Bring radical.  There is a good timeline at Wolfram's site.
