How many times have the cells in a human body divided?
The supplemental material of the paper that Dirigible references in a comment to my question gives a rough estimate of these values. If I understand their simple model correctly, then we can estimate the average number of times that the stem cells in various types of tissues have divided over an 80-year lifespan from Table S.1 by finding the row corresponding to a cancer of that tissue, and then calculating $\log_2(s) + d$, where $s$ and $d$ are given in the table. (For specialized cells, I guess you'd add 1 more to capture the final cell division that produces the specialized cell?)

I haven't done the math out, but glancing over the table, it looks like the generation number varies widely between tissues, from a low of about 27 for some brain cells to a high of almost 6000 for cells in the colon and rectum.