The totally honest truth is that the periodic table doesn't actually end.
I mean it stops for now, but it isn't finally finished because we keep making elements. The naturally-occurring elements stopped around 92 with uranium (having skipped technetium and promethium on that journey to uranium). Since then, we've only found the rest of the 118 (for now) elements via particle accelerator, bombarding naturally occurring elements with other naturally occurring elements.
Sam Kean - author of The Disappearing Spoon, a great chemistry book - goes through some of the details of the man-made elements here in sort of answering the question about 'where does the periodic table end?'
Though as Eric Scerri wrote a few years back...
It is simply not yet clear whether the principle that elements in the same column in the periodic table behave similarly remains valid for very heavy atoms. The question is of no great practical consequence, at least for the foreseeable future. The loss of predictive power in the superheavy realm will not affect the usefulness of the rest of the table. And the typical chemist will never get to play with any of the elements of highest atomic numbers: these elements' nuclei are all very unstable, which means that they decay into lighter elements instants after being created.
Still, the question of special relativity's effect strikes at the very heart of chemistry as a discipline. If the periodic law does lose its power, then chemistry will be in a sense more reliant on physics, whereas a periodic law that holds up would mean the field maintains a certain level of independence. In the meantime, perhaps, Mendeleev's ghost should just kick back and marvel at the success of his favorite brainchild.
