Which NP-Complete problem has the fastest known algorithm?
Vertex Cover has an algorithm running in time $1.2738^k + nk$, and is thus faster than $2^n n^2$, even with $k=n$.  You can check out Table of FPT races for a short list of FPT running times of different problems. Here, $n$ is the number of vertices and $k$ is the solution size.

Also, the question Are there subexponential-time algorithms for NP-complete problems? addresses similar questions.