Algorithms for set packing There seems to be much work, for some NP-Hard problems, on developing fast exponential time exact algorithms (i.e., results of the form: Algorithm A solves problem $x$ in O(c^n) time, with c small). There seems to be a fair amount of work along these lines for some NP-hard problems (e.g., Measure and conquer: a simple $O(2^{0.288n})$ independent set algorithm. SODA’06) but I haven't been able to find similar work for the set packing problem. There seems to be similar work on some restrictions of the set packing problem (e.g., An $O^{*}(3.523^{k})$ Parameterized Algorithm for 3-Set Packing) but I haven't found any for the general set packing problem.

So my question is: What is the best time complexity for exactly solving the weighted set packing problem where there are $m$ sets drawn from a universe of $n$ elements?

I am also interested in the relationship between the number of sets and the size of the universe. For example, has there been algorithmic work on situations where $m$ is relatively large compared to $n$ (i.e., close to $2^n$)?
Indeed, set packing, partitioning, and covering have been studied in terms of exact algorithm running times.
 To address your last question, you can solve weighted set packing in $O(m2^n)$ time by dynamic programming across all subsets of $[n]$.
 Moreover, if your integer weights are bounded by $M$, you can solve it in $O(M2^n)$ time, even if $m$ is as large as $2^n$, see

http://dx.doi.org/10.1137/070683933

BTW, The parameterized result you list for $3$-sets is not the best known, see

http://arxiv.org/abs/1007.1161

for a state-of-the-art algorithm and a list of previous results on the problem.