Good question!
Theoretically, there is no upper limit (although see note below on mass limits in neutron stars), but elements become less and less stable as you get heavier and heavier. The heaviest element in existence that is always stable is lead - anything heavier than that is radioactive, and (in general, this isn't absolute) as you continue getting heavier, the elements have shorter and shorter half-lives. It has been speculated that there is another "island of stability" around atomic numbers of 150-200 or so, but that it's virtually impossible to get that high because so many of the intermediate steps are so short lived. Whether this is true or not is at the moment still just a theory with no data to support or disprove it.

Now, that said, the stability of elements in the periodic chart is determined by the dominance of the weak and strong nuclear forces. There are objects in the universe called neutron stars which form in the supernovae of massive stars and that are so dense and so massive that their gravity is powerful enough to all but overcome the nuclear forces, holding them together at nuclear densities and preventing them from decaying. Neutron stars can have masses upwards of a few times the mass of the sun - we don't know exactly how far, because there is the possibility of another state in which the nucleons (protons and neutrons) collapse to form a giant nucleon that has been referred to as a 'quark star', although quark stars have not been observed. Above five or so solar masses, there is a genuine upper limit, because gravity becomes so strong that the nuclear forces utterly give way, and the whole thing collapses to form a black hole. There is no limit to how big they can get.

The primary obstacle to creating massive atoms is the binding energy pernucleon (protons and neutrons) decreases as atomic mass increases. Generally speaking, the nuclear strong force, mediated by gluons, isattractive only at very, very small distances, while electrostatic forcesare repulsive for like charges.Since the protons are positively charged,they repulse each other. As the nucleus gets larger due to having morenucleons, the strong force attraction is not as powerful for nucleons thatare distant from each other. Eventually, the strong force cannotcompensate for the electrostatic repulsion when enough protons are addedand the nucleus gets very large.