Why are there only 7 periods in the periodic table?
There is no theoretical limit to the number of rows, but...

On one hand, as Tom Lehrer states, there may be many others but they haven't been discovard.
On the other hand, after all s, p, d, f orbitals are filled in the seventh period, there might be a new row, or there might be an extension of the seventh row, as a new type of orbital (g? Anyone know it there's a name?). Each orbital has double an odd number (double because the Pauli Exclusion Principle allows pairs of electrons of opposite spins), so the next orbital would have 18 electrons. Is it possible? Inner electrons would be moving at very high speeds, and they'd also shield outer electrons from attraction by the positive nucleus.
On the gripping hand, transuranics may last a few seconds... or a few attoseconds, so getting spectra might be problematic.