Claudiu, the topic of ultrafinitism is quite slippery, and by no means well-defined (unlike other more conservative forms of constructivism, such as intuitionism, Bishop's constructivism, and so on), at least not yet.
I suggest that you go to FOM, and google ultrafinitism, to have an idea of what some folks think about the subject. For instance, some "ultrafinitists" are ACTUALISTS, meaning that they believe in some physical max integer number, whereas others (such as myself) are not.
I do not believe in ANY number (in the ontological sense), not just huge ones. In my perspective, which I think could be better denoted characterized as ULTRAFORMALIST, syntax is all you have, and "numbers" are simply the by-products of playing the arithmetic game according to the arithmetical rules + basic logic.
There is, though, or so it seems to me, a common thread among ultrafinitists, namely the general mistrust for every form of infinitary reasoning, included so-called potential infinity (both at the mathematical and meta-mathemathical level).
Now, from my perspective, as Qiaochu already has already observed, $\pi$ does exist, in the sense that there is a very finite object (the algorithm), to compute it, at any degree of precision you like (provided, of course, that you have the computational resources to carry out the computation). Think of a computer program: your laptop "understands" what $\pi$ is, and then it simply produces its digits. At some point it will stop, because it runs out of gas. Your formula has to be emended so that it says: IF you can compute up to some N, you can compute $\pi$ accordingly with the given precision.
You question can be generalized to a much broader one, namely what exactly is the viewpoint on mathematical objects at large from such a perspective. I think one day (in my opinion quite soon) we will have such a viewpoint fully articulated, and it will look somewhat like Brouwer's theory of species, only without giving free tickets to any type of infinitary reasoning.
ADDENDUM: even actualists, for instance Doron Zeilberger (i hope I understand him correctly), would accept your PI, only they would argue that its status is symbolic, as opposed to concrete numbers (say 5). They have no problem in manipulating symbolic numbers, they simply do not think they correspond to anything "out there".