Can Uranium-235 absorb neutrons and become U-238 instead of splitting? Does that happen, on occasion?  Does anyone know?
I suppose it is quite rare, ... To absorb three neutrons without once fissioning...
Could U-235 absorb three neutrons at once?  Or just one, becoming U-236, then two at once?
 A: Three neutrons at once, no.  But we can look at the probabilities for each step.
Using these cross sections for "thermal" neutrons with energy 25 milli-eV, the total cross section for (n, fission) is about 460 barns, while the cross section for (n, gamma) (that is, capture to U-236) is about 60 barns.  So for every seven or eight fissions, you have one capture to U-236.
For U-236, the cross section for neutron capture (0.3 barn) is much larger than the cross section for fission (0.0028 barn).
For U-237, the cross sections for capture (0.16 b) and fission (0.15 b) are about the same.
So the answer is that, in a reactor that is fissioning a chemically significant amount of uranium, you'll transmute plenty of U-235 to U-238.  The bottleneck is probably U-237, which has a relatively low capture cross section and a short (week-ish) half-life.  The fraction of U-237 that beta-decay to neptunium (which has its own different capture/fission cross sections) are part of the route to plutonium enrichment.
