The recent discoveries of ununtrium and ununpentium made me realize that I hadn’t seen a table of elements in a while. Curosity led me to Wikipedia’s extended ToE which lists (mostly projected) elements to 218.
Is there a limit to the number of elements that can exist, and why?
Mostly projected? Mostly projected??? My dear evilray, a mere 105 out of 218, or 48.2%, of the elements on that table have not yet been created and confirmed. 
Bubba. You’ve asked a mouthful.
The energies involved in pushing more mass and energy into an existing massive nucleus are literally astronomical. My guess is you gonna die before anyone sees ony of those long-winded elements come to fruit.
No worries, though. I’ll probably go first…
I think I read somewhere that the max theoretical atomic number was 137. Anything heavier, and electrons have to go faster than the speed of light or they fall into the nucleus (reverse beta decay) and you get another neutron. I guess this has to do with the fact that the likelihood an electron absorbs a photon is related to the inverse, 1/137. When you’ve got 137 protons, the likelihood an electron will interact with it becomes about 100%.
However, I’ve also read that somewhere in the 120s, protons will be so far apart that eletromagnetic forces could overwhelm the strong force (which is very short range, as it involves the exchange of massive mesons called pions), and hence the thing will simply decay instantaneously, probably by emitting an alpha particle or something like that.
Of course, I could be completely full of shit here, as I’m not a physicist!
It depends on what you mean by “exist.”
The majority of the elements listed on the extended table either (a) have never been observed or (b) are hideously unstable. There’s good reason for that, too, which I’ll get to in a minute. But first, I want to emphasize that if by “exists,” you mean “can be observed and has some non-zero lifetime,” then I don’t think that there’s any limit in principle to the number of elements. If you want the lifetime to be non-negligible, on the other hand…
Now, the reason that elements with higher atomic numbers are unstable has to do with the relative ranges of the strong force (which holds nuclei together) and the electromagnetic force (which tends to push nuclei apart). The strong force is stronger (hence the catchy name), but vastly shorter in range. So what happens as you go to large atomic number is that the protons are only being bound by the strong attraction to a few neighboring nucleons, while they are being pushed away by all the other protons. Eventually, you can’t get enough nucleons to overcome the Coulomb repulsion between the protons, and the nuclei wind up not being stable.
On preview, Loopydude has said something similar already. I want to point out that in QM, the electron doesn’t have a well-defined velocity… However, the average velocity for an electron near a nucleus of charge Z is in the ballpark of Z/137 c, albeit a bit lower. But this is for small Z, as it comes from non-relativistic QM, and clearly by the time the average velocity of the electrons nears the speed of light, we can’t use non-relativistic theory anymore.
Yah. I think I remember that these relativistic effects become unignorably important once you hit the lanthanide series; electrons are moving so fast around these big ole’ nuclei that their mass is huge and they orbit very close to the nucleus. Hence the “lanthanide contraction”.
So, in relativistic QFT, when do electrons have to moving, on average, of course :), at c in order to resist inverse beta decay?
Haven’t a clue, actually. Beta decay is not one of those things I really know an awful lot about.