  We address the problem of computing bounds for the self-intersection number
(the minimum number of self-intersection points) of members of a free homotopy
class of curves in the doubly-punctured plane as a function of their
combinatorial length L; this is the number of letters required for a minimal
description of the class in terms of the standard generators of the fundamental
group and their inverses. We prove that the self-intersection number is bounded
above by L^2/4 + L/2 - 1, and that when L is even, this bound is sharp; in that
case there are exactly four distinct classes attaining that bound. When L is
odd, we establish a smaller, conjectured upper bound ((L^2 - 1)/4)) in certain
cases; and there we show it is sharp. Furthermore, for the doubly-punctured
plane, these self-intersection numbers are bounded below, by L/2 - 1 if L is
even, (L - 1)/2 if L is odd; these bounds are sharp.
