He worked on both ultrafinitism and nonstandard analysis, bringing new approaches to both. He was discussed around here a couple of years ago for his claim to have proved the existence of a contradiction in Peano Arithmetic, but Terry Tao found the flaw. Unfortunately, he died last year.

In his nonstandard days, he wrote Radically Elementary Probability Theory. This is not simply redoing standard continuous probability theory with nonstandard analysis, but doing discrete probability theory with nonstandard integers. All of the useful things appear, but it all looks very different. Princeton still hosts the PDF: https://web.math.princeton.edu/~nelson/books/rept.pdf

ETA: While ultrafinitism and nonstandard analysis seem almost complete opposites as far as constructivism goes, his approaches to them are actually quite similar to one another.