My question is basically motivated by the fact that I prefer to create images using metapost - as a consequence, all TeX processing of formulas/labels inside of images is done there. After that, pdflatex eats those images just fine. (Graphicx package etc.)

Could you please clarify what you mean by label placement inside of figures?

Wow. I think this is the first time I regret that I can only vote once for an answer.

@Barry: fair enough, however my personal belief is that prime numbers belong to the list of most basic concepts of number theory, and therefore the use of their infinitude in a proof of any number-theoretic result is fully justified.

Ah okay, so the property of Gamma that is crucial is that for $g(t)=Gamma'(t)/Gamma(t)$ satisfies $g(t+1)-g(t)=h(t)$ where $h$ is a rational function (some additional properties $h$ has to have are in the paper, just wanted to extract a highlight).

Why would you want to avoid primes? Avoiding number theory at all costs when talking about algebraic/transcendental numbers feels very close to walking to work on your hands...

sure, I can add this tag

Thanks for the links! "is it true that "natural" generating functions are either differentially finite or differentially transcendent?" I would not think so - see the sec(t) example mentioned above - if that is not "natural" enough, think of up-down permutations...

Wadim: sure, I was rather curious about what are simple and natural methods to check it for a function. I guess two methods are explained in answers below: that Thue-Siegel-Roth-type theorem and the p-adic approach. I don't have Gelfond's book in proximity. Could you give a hint on how Gamma is handled?

In case someone is curious about the method: the authors refer to the paper MR0604044, Sibuya, Yasutaka; Sperber, Steven, Arithmetic properties of power series solutions of algebraic differential equations. Ann. of Math. (2) 113 (1981), no. 1, 111--157. There for a series whose coefficients are algebraic numbers it is proved that if it is differentially algebraic, then it is convergent in some neighborhood of zero w.r.t. every non-Archimedean valuation.

Thanks a lot, that was really helpful! My search brought up an even stronger result, in the spirit of Thue-Siegel-Roth: MR048568 Osgood, Charles F. Concerning a possible ``Thue-Siegel-Roth theorem'' for algebraic differential equations. Number theory and algebra, pp. 223--234. Academic Press, New York, 1977.

I updated the original question with an additional one motivated by your first example, and an example from the other comment. Do you by any chance know how the proof for Gamma goes? What intuition of Gamma should one use?

Sounds like a plan indeed. This example, as well as the example in the other comment (with $2^n$'s - it probably works for a similar reason) makes me think of all those Liouville-flavoured examples of transcendental numbers, - I think I might make an update to my question with another question arising from that!

I believe that people do not have to justify their aesthetic preferences, - and if you are curious to know why they liked this problem, asking it in a comment to the original question would seem much more appropriate to me!