The Warren Buffett Problem
I imagine there are lots of possible approaches to this problem (many of which I'm sure you considered) -- here are a few ideas/references.


You could play this as $N$ independent parallel single-arm bandit games, deciding to pull or not pull each arm independently.  This should work especially well if the rewards are independently distributed.
Allow each set of arms to be a new arm and run an Exp3-type algorithm.  This gives a $O(2^{N/2} T^{1/2})$ regret - not so great.
In an upcoming NIPS 2010 paper, Saten Kale, Rob Schapire, and I consider the case where one plays a slate of arms at once.  In our work, however, the size of the slate is fixed.  This paper also considers a similar problem.  Another similar work appeared in ALT 2010.  Perhaps some of the ideas transfer.
If you treat it as an experts problem (each expert recommends a different one of $2^N$ subsets), by following one expert, you can get estimate the performance of other experts who have non-empty intersections in their choice of arms to pull by using importance weighting.  An Exp4 type analysis might get you $O(N\sqrt{T})$ regret but $O(2^N T)$ running time.


EDIT below: 

It seems to me that the budget constraint (not going below $0$) makes the problem intractable.  Imagine you have a budget of $1$.  The adversary can make one of the  arms always pay off and the rest never pay off.  So w.p. $(n-1)/n$ you go bust in the first round while the optimal strategy gets $T$ dollars after $T$ rounds.  So your expected regret is at least $(n-1)T/n$ and you can't hope for a high probability bound at all.

It also seems that this can work for any initial budget.  Say you start with $B$ dollars.  Then the adversary can set all but one arms to pay $0$ and one arm to pay something like $2B$ w.p. $1/B$.  I guess if you have a limit on the possible payout amount and a high enough initial budget, then this might leave room for an interesting problem.