[Matthew Parkinson told me this problem, or a slight variation thereof.]

You are given a balance (that is, a weighing machine with two trays) and a
positive integer N. You are then to request a number of weights. You
pick which denominations of weights you want and how many of each you want.
After you receive the weights you requested, you are given a *thing*
whose weight is an integer between 1 and N, inclusive. Using the balance
and the weights you requested, you must now determine the exact weight of the
*thing*.

As a function of N, how few weights can you get by requesting?

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