Let $k$ be a field. What is an explicit power series $f \in k[[t]]$ that is transcendental over $k[t]$? 


I am looking for elementary example (so there should be a proof of transcendence that does not use any big machinery).



What's an example of a transcendental power series?
If $k$ has characteristic zero, then $\displaystyle e^t = \sum\_{n \ge 0} \frac{t^n}{n!}$ is certainly transcendental over $k[t]$; the proof is essentially by repeated formal differentiation of any purported algebraic relation satisfied by $e^t$. 


**Edit:** Let me fill in a few details. Given a polynomial $P$ in $e^t$ of degree $d$ where each coefficient is a polynomial in $k[t]$ of degree at most $m$, the possible terms that appear in any formal derivative of $P$ lie in a vector space of dimension $(m+1)(d+1)$, so by taking at least $(m+1)(d+1)$ formal derivatives we obtain too many linear relationships between the terms $t^k e^{nt}$. The coefficient of $e^{dt}$ in particular eventually dominates all other coefficients.


