TITLE: An example of a series that is not differentially algebraic? 
QUESTION [14 upvotes]: Motivated by this question, I remembered a question I was curious about sometime which I am sure has some easy and nice example for it as well, which I just can't think of for some reason. I want an example of a power series that is not differentially algebraic. A differential algebraic power series is a series $f(t)$ satisfying an equation $P(t,f(t),f'(t),\ldots,f^{(k)}(t))=0$ for some $k$ and some polynomial $P$ in $k+2$ variables. 
Update: examples in the comments below ($\sum t^{n^n}$, $\sum t^{2^n}$) make me ask a refinement (of a sort) for the original question: these examples are reminiscent of all those Liouville-flavoured examples of transcendental numbers, - I wonder if there is a Liouville-flavoured proof, stating that if the polynomial P is of given (multi)degree, some inequality holds that is obviously impossible for the series above?
Update 2: there are quite a few examples now, and I am tempted to accept the $\sum t^{n^n}$ answer since the example itself is easy and it came together with an easy explanation. I wonder what are other general approaches besides the ones that are exhibited in answers here (looking at p-adic norms of coefficients and looking at powers of $t$ with nonzero coefficients).

REPLY [4 votes]: Yet some more examples: the ordinary generating function for the Bell numbers, see Martin Klazar.  Or, Flajolet, Gerhold, Salvy.
I think it would be nice to have some kind of survey of results and methods.  In combinatorics, I frequently encounter generating functions that do not seem to be differentially algebraic, but I have no idea how to prove that.  An example would be the generating functions for walks in the quarter plane with certain step sets, see Bostan, Kauers.  Maybe a little bit more philosophical: is it true that "natural" generating functions are either differentially finite or differentially transcendent?