TITLE: Number of unlabelled planar graphs
QUESTION [8 upvotes]: What are the best known bounds on the number of non-isomorphic (unlabelled) planar graphs on $n$ vertices? Is there a simple proof that this number is at most exponential in $n$?

REPLY [2 votes]: There is a short proof of a stronger result by Norine, Seymour, Thomas, and Wollan that there is an exponential upper bound for every proper minor-closed class.

For every proper minor-closed class of graphs $\mathcal G$, there exists a constant $c$, such that for every integer $n$, there are at most $n! c^n$ graphs in $\mathcal G$ with vertex set $\{1, \dots, n\}$.  

This is the labelled version of your question, but as Igor mentions, the typical graph has trivial automorphism group, so we can divide by $n!$.  
The paper is entitled Proper minor-closed families are small and can be downloaded here or from Robin Thomas's webpage.