Taken literally, nearly 2,500 people have run for president since 1980, in the sense that they filled out the requisite form with the Federal Election Commission (which is remarkably trivial to do). Since many have run multiple times, it's a grand total of more than 3,100 separate candidacies.
For elections that have produced a total of five different presidents.
(Attention graduate students: Is the increase in candidates in the post-Internet world a function of the Internet? Please submit your research to Philip Bump c/o The Washington Post.)
You'll notice that we have differentiated between "real" and non-real candidates -- a very blurry distinction. It's commonly accepted that the "full" 2016 Republican field will have 16 candidates, but that leaves out, say, Mark Everson, whose credentials are not any worse than some of the 16 magic '16ers.
In putting this graphic together, I tried to answer the question objectively: What makes someone a "real" candidate. Andrew Lessig is not a real candidate. Jeb Bush is a real candidate. Where's the middle ground? What's the boundary?
We tried looking at the number of campaign committees for each candidate. The more campaign committees -- Iowa for Obama, etc. -- the more real the candidate, it would seem, no? It's not a bad proxy, but it means that candidates like Billy Joe Clegg would have counted as "real" in 1996.
Instead, the figure above uses a more subjective objective definition: Appearances on Wikipedia. Let the wisdom of the crowd tell us who is considered real or not! And since Wikipedia includes some of the fringier members of the fringe, it's not totally unfair to those who garnered some attention.
In the truest, most objective sense, the Republican 2016 field is not the largest in recent memory. The 2012 field was larger. At least so far. Still plenty of time for more non-starter Republicans to jump into the race.