Ultrafinitism is basically resource-bounded constructivism: proofs have constructive content, and what you get out of these constructions isn't much more than you put in.
Looking at the universal and existential quantifier should help clarify things. Constructively, a universally quantified sentence means that if I am given a parameter, I can construct something that satisfies the quantified predicate. Ultrafinitistically, the thing you give back won't be much bigger: typically there will be a polynomial bound on the size of what you get back.
For existentially quantified statements, the constructive content is a pair of the value of the parameter, and the construction that satisfies the predicate. Here the resource is the size of the proof: the size of the parameter and construction will be related to the size of the proof of the existential.
Typically, addition and multiplication are total functions, but exponentiation is not. Self-verifying theories are more extreme: addition is total in the strongest of these theories, but multiplication cannot be. So the resource bound is linear for these theories, not polynomial.
A foundational problem with ultrafinitism is that there aren't nice ultrafinitist logics that support an attractive formulae-as-types correspondence in the way that intuitionistic logic does. This makes ultrafinitism a less comfy kind of constructivism than intuitionism.
Why do people believe it? For the same kinds of reasons people believe in constructivism: they want mathematical claims to be backed up by something they can regard as concrete. Just as an intuitionist might be bothered by the idea of cutting a ball into non-measurable pieces and putting them back together into two balls, so too an ultrafinitist might be concerned about the idea that towers of exponentials are meaningful ways of constructing numbers. Wittgenstein argued this point in his "Lectures on the Foundations of Mathematics".
Can you actually get math done from that perspective? Yes. If intuitionism is the mathematics of the computable, ultrafinitism is the mathematics of the feasibly computable. But the difference in ease of working with between ultrafinitism and intuitionism is much bigger than that between intuitionism and classical mathematics.
In the formal meaning of "computable" the floor of that number is indeed computable. This is to say that a patient immortal human with access to unlimited paper and pencil could, in principle, work out the answer. (Here I assume, for technical reasons, that the number in question is not an integer - I assume someone who knows enough number theory will be able to cite a result that implies this.)
The article linked makes the weaker claim that the value has not yet been calculated, which seems likely to me. The issue they are concerned about is that humans are not immortal and that our supply of paper is very limited. If the number of decimal digits in the value is too large, it would be impossible to actually represent it in any physical way within our universe.
In general, I think it is more accurate to say that ultrafinitists don't accept that the set of all natural numbers is a coherent entity - not that they think it is finite. However, as the article you linked alludes, it is very difficult to find a coherent but non-arbitrary way to say what natural numbers are without accepting that there are an infinite number of them.
Addendum Here is why I am worried whether $e^{e^{e^{79}}}$ is an integer. It's certainly correct that no matter what, the floor of that number is an integer and is therefore computable. That part of my argument is fine.
On the other hand, if $e^{e^{e^{79}}}$ is not an integer, then I can tell you a specific algorithm to use to compute it. Namely, compute better and better upper and lower bounds until they fall strictly between two consecutive integers (which they must, since their limit is not an integer) and then pick the smaller of those two integers.
If $e^{e^{e^{79}}}$ is an integer, then that algorithm won't work, because it will never stop. But if we knew that $e^{e^{e^{79}}}$ was an integer then we could take better and better upper and lower approximations until they straddle a single integer, and then pick that.
So the reason that I am interested whether the number is an integer is that, beyond merely knowing that the floor is an integer, I'd like to know which algorithm could be used to compute it.
In any case, I don't think that the point of the example was to pick a number that is not known to be integer or known to not be an integer. The point of the example should be to pick a number which is simply too large to represent physically. I was hoping that someone would have a quick answer that confirms $e^{e^{e^{79}}}$ is not an integer, so I could edit my response with that info. But the non-integer property seems more difficult than I thought.
Best Answer
It is simply that there is no formal system that is widely accepted as representing (a least common denominator of) ultrafinitist ideas. For constructivism, we have things like intuitionistic propositional logic, Heyting arithmetic, and constructive set theories. If you work in these systems you will generally be doing things that no constructivist will complain about. Similarly for predicativism (e.g. the internal language of pretopos) and finitism (PRA).
There are formal systems that are presented as "ultrafinistic", e.g. the system sketched in Sazonov's On Feasible Numbers, but I doubt Sazonov had any intent for this to be a comprehensive system that should be widely adopted by ultrafinitists. Vopěnka's Alternative Set Theory seems like it was intended as a more serious proposal, but (from the little I've read about it, so take this with a large grain of salt) it seems more focused on the notion of semisets (subclasses of sets) which provides an approach to vague predicates, e.g. the kinds of things that come up in the Sorites paradox. The relevance to ultrafinitism is that we can assert that all sets are "hereditarily finite" (or, rather, satisfy a first-order induction schema that roughly corresponds to that) but then assert that there's a proper semiset to model the "humanly describable" sets. The more levels of nested sets we exhaustively enumerate, the deeper this proper semiset must be but it is still contained in some set (and all sets are "hereditarily finite"). This allows reasoning about a "cutoff" "somewhere" but somewhere that's always beyond the horizon. However, this doesn't explicitly provide any concrete upper bound. I'm not familiar enough with Alternative Set Theory to say for sure, but I suspect that you could probably define some concretely very large things that would bother ultrafinitists.
Another aspect which is part of the point of Sazonov's paper is that an ultrafinitist theory would need an ultrafinitist meta-theory. Proof theory lets us reduce most other theories (e.g. ZFC) to a finitist meta-theory. You don't need to believe in uncountably infinite sets to believe in manipulating syntax that purports to talk about them in some deductive system. However, "reducing" an ultrafinitist theory to a finitist meta-theory isn't much of a reduction (which doesn't mean it wouldn't be useful). There are also aspects of the meta-theory that can leak back in to the theory.