An example of a series that is not differentially algebraic?  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).
 A: You'd be better off in characteristic zero, for $f^{(p)}(t)=0$ in characteristic $p$. Then the sea of zeroes example $\sum_n t^{n^n}$ will do the trick. For large enough $n$, there will be a cluster of non-zeroes in degrees $kn^n-m$ for small (and bounded) $k$ and $m$, "reachable" only by products of $(t^{n^n})^{(s)}$, the same $n$, bounded $s$. Their vanishing will give infinitely many linear relations on the coefficients of $P$, which you can explicitly write down and see that there are no nonzero solutions on coefficients.
A: In Exercise 6.63(c) of Enumerative Combinatorics, vol. 2, I raise the question of whether one can ever have   $\sum b_i x^{n_i}$ differentially algebraic (DA) (over $\mathbb{C}$) if $b_i\neq 0$ and $\lim_{i\to \infty}i^2/n_i=0$. In particular, is $\sum x^{n^3}$ DA?
A: 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?
A: I've always seen the canonical example for this to be $\sec x$. For this and more examples, see R. Stanley's excellent article on the subject, "Differentiably finite power series" 
Oops, above I am referring to D-finite, power series, but you are referring to D-algebraic power series. It is proved in "A gap theorem for power series solutions of algebraic
differential equations" by L. Lipshitz and L. Rubel that 
$$\sum_{n=0}^{\infty}x^{2^n}$$ is not D-algebraic.
Another function that was proven not to be D-algebraic is the Gamma function, and this fact is due to Holder.
A: C. Osgood proved a Liouville type theorem for algebraic differential equations.
A: An example is given by
$$f(t)=\sum_{n=1}^\infty \frac{(n-1)!}{n^n}t^n$$
see http://arxiv.org/abs/math.CA/0210472
