Nelson (and at the linked question, we see also Doyle and Conway) is famous for not believing in the existence of $\aleph_0$, which is the cardinal of a limit ordinal (i.e. $\omega$), so can be considered large, but not large from a usual set theory point of view. He would be called an ultrafinitist a finitist in this respect. More precisely, his axioms of arithmetic do not presuppose the existence of a natural numbers object, and do not show it either.
Edit: In fact Nelson is an ultrafinitist, in that he doubts the existence of large natural numbers, and gives a combinatorial example (see my comment below) of a number the finiteness of which he questions (this corrects a mis-statement on my behalf on the original version of the question, where I called Nelson an ultrafinitist for not 'believing in' $\aleph_0$).