Element 137 - Theoretical limit I've seen a few sources that mention the speed of an electron in a hydrogen atom is 1/137 the speed of light. This article also mentions what looks like a correlation between atomic number and the numerator meaning gold's electron speed is 79/137 the speed of light.
Does this imply that the highest element possible would be element 137 (or more likely 136)? Not referring to anything stable, just what is possible with physics (and hadrons).
 A: No.
There is indeed a correlation between the nuclear charge $Z$ of an element and the kinetic energy of the innermost electrons, but there are two things to keep clear sight of:

*

*The correlation is not linear, so you cannot simply multiply the speed of the innermost electron by $Z$ and hope for things to just work out. The interior of a large atom is a very complex place, with many dynamical effects going on, and even if the kinetic energy of hydrogenic orbitals does increase with $Z^2$ that does not mean that this will remain true when multiple electrons are present. To begin with, even in helium the average kinetic energy of either electron is not four times that of hydrogen - so there's that settled.


*More importantly, it is the kinetic energy that increases with $Z$. Once the corresponding electron speed approaches the speed of light, you have to switch over to the relativistic expression of kinetic energy,
$$E_K=\frac{mc^2}{\sqrt{1-v^2/c^2}}-mc^2,$$
and this can increase without bounds while still maintaining $|v|<c$. That is, you can keep adding energy to the electron without fearing that it will break the speed of light, so your fears are unfounded.
Going a bit further, the article you linked to is fairly misleading on several fronts. The presence of relativistic effects inside atoms has been perfectly appreciated, from the fine structure of hydrogen to strongly relativistic effects in heavier atoms like the "anomaly" of mercury being liquid, which are bread and butter to the atomic physicists that study them.
Nowadays, and for the past sixty years or so, having atoms "deviate from their expect chemical behavior predicted by their place on the Periodic Table of Elements" simply means that the calculations were either taken naively or without sufficient care. The behaviour described (the decoupling of the S orbitals of Sg from the valence shell due to relativistic energy considerations, leaving an effective valence of +4 instead of the naive +6) is in no way unprecedented.
