Over on Google+ I wrote:

Ultrafinitists don’t believe that really large natural numbers exist. The hard part is getting them to name the first one that doesn’t.

Richard Elwes replied, pointing out a conversation between the ultrafinitist Alexander Esenin-Volpin and the master of large infinities, Harvey Friedman. Friedman challenged Esenin-Volpin to name the first number that doesn’t exist:

I raised just this objection with the (extreme) ultrafinitist Esenin-Volpin during a lecture of his. He asked me to be more specific. I then proceeded to start with 2 1 and asked him whether this is “real” or something to that effect. He virtually immediately said yes. Then I asked about 2 2, and he again said yes, but with a perceptible delay. Then 2 3, and yes, but with more delay. This continued for a couple of more times, till it was obvious how he was handling this objection. Sure, he was prepared to always answer yes, but he was going to take 2 100 times as long to answer yes to 2 100 then he would to answering 2 1. There is no way that I could get very far with this.

I believe that after each question Esenin-Volpin was mentally counting to the desired power of 2, to verify that it really exists.

(In other words, he was checking that it could be constructed by repeatedly applying the successor operation to the number 0.)