What type of distribution does this most closely resemble? I was thinking that this graph didn't really resemble any type of distribution.
This appears to come from https://rdrr.io/rforge/Lahman/man/Pitching.html, so sports data. One row per player per season since 1950. GS is games started.
This distribution probably doesn't have any nice name, because the distribution itself is not nice. That doesn't mean that it doesn't follow any distribution. Just describe it in some useful qualitative terms: it's bimodal, with approx 25% of the sample at 0, and a small local mode at 33. The max is (approx) 36. Really, the histogram (more of a barplot actually) tells the whole story.