See David Speyer's thread
http://mathoverflow.tqft.net/discussion/741/does-lim-cosn-exist/
which is about a closed question involving limit points of $\cos (n!)$ which is to say values of $n! \pmod {2 \pi},$ or in turn values of $$ \frac{n!}{2 \pi} \pmod 1. $$ David's conclusion, evidently posted on math.stackexchange, is that we do not know enough about $\pi$ to answer this.
