Formal power series with radius of convergence 0 often arise in counting labeled graphs. For example, the exponential generating function for labeled connected graphs is $\log G(x)$, where $$G(x) = \biggl(\sum_{n=0}^\infty 2^{\binom{n}{2}} \frac{x^n}{n!}\biggr),$$ which has radius of convergence 0. 

As Aaron noted, series like $\sum_{n=0}^\infty n! x^n$ arise in the theory of continued fractions;  this series has the continued fraction expansions
$$
\frac{1}{1-\displaystyle\frac{\mathstrut x}{1-
\displaystyle\frac{\mathstrut x}{1-
\displaystyle\frac{\mathstrut 2x}{1-
\displaystyle\frac{\mathstrut 2x}{1-
\displaystyle\frac{\mathstrut 3x}{1-
\displaystyle\frac{\mathstrut 3x}{1-\cdots
}}}}}}}
$$
and
$$
\frac{1}{1-x-\displaystyle\frac{\mathstrut x^2}{1- 3x -
\displaystyle\frac{\mathstrut 2^2x^2}{1-5x -
\displaystyle\frac{\mathstrut 3^2x^2}{1-7x -\cdots
}}}}
$$


Similar continued fractions exist for ordinary generating functions (with radius of convergence 0) for Bell numbers, Eulerian polynomials, matchings, and more generally, moments of orthogonal polynomials. A very nice combinatorial approach to these continued fractions has been given by Philippe Flajolet, [Combinatorial aspects of continued fractions][1].

It is true that most, if not all, of these examples of nonconverging power series can be refined to power series in more than one variable that do converge for some values of the parameters. For example, the exponential generating function for labeled connected graphs by edges is $\log G(x,t)$, where
$$G(x,t) = \biggl(\sum_{n=0}^\infty (1+t)^{\binom{n}{2}} \frac{x^n}{n!}\biggr);$$
this converges for $|1+t|<1$. On the other hand, the exponential generating function for strongly connected tournaments is $1-1/G(x)$, and this doesn't seem to generalize since $1-1/G(x,t)$ has some negative coefficients.




  [1]: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.74.6909&rep=rep1&type=pdf