I'm answering my own question for the benefit of anyone else who might be wondering about this. (Please note that I'm just a programmer, not a biologist, but thanks to David's comment, I was able to come to what I think are some correct conclusions.)
A single mRNA strand can be translated more than once, even more than once at the same time. The
polysome that David references is a group of two or more ribosomes all attached to the same mRNA strand, all working on translating it simultaneously.
I believe that it may be possible for a single mRNA strand to be translated hundreds or even thousands of times. My reasoning for this is that a virus injects one copy of its RNA into a cell, and from this one copy, new viruses are replicated until the cell bursts. This "burst size" varies by host cell and virus, but it can range from hundreds to tens of thousands or more. While there are still a lot of things I don't understand about the specific steps the RNA goes through (are there intermediary copies made from which multiple mRNA strands are produced?, etc.), it still seems to suggest that a single strand of mRNA can be translated a lot.