What are the consequences of having an element with an atomic number above 137? In other words, is an element such as Feynmanium the last chemical element that can physically exist on a planet such as Earth?
The critical value &quot;137&quot; occurs in calculations based upon relativistic theories that treat the nucleus as a point, such and Dirac theory or Sommerfeld theory.
However, when the nucleus is correctly treated as having a non-zero size, a significantly higher value is calculated.
The most recent calculations I see are in New method for solving the $ \bf Z &gt; 137$ problem and determining hydrogen-like energy levels (2014) Phys.-Usp. 57, 189.  (alternative link to pre-print):

Following the development of the Dirac theory in 1928, Dirac [reference 2], Darwin [reference 3] and Gordon [reference 4] obtained expression (1) as a result of exact solution of the Dirac equation in the Coulomb field of a point charge (-Ze). ...  Formula (1) [results in a complex number] if Z [is greater than 137]...the complexity of energy levels in (1) is often called the “Z&gt;137 catastrophe”.
In 1945, Pomeranchuk and Smorodinsky [6] considered
an atomic system with [the nucleus being a sphere of radius $0.8 \times 10^{-12}$cm and having uniform charge within the nucleus and found 175 to be the critical value].

The article goes on to explain that since the 1945 calculation, various calculations, with various nuclear charge distribution models, have calculated critical values in the 170-175 range.  The article's own calculations find 178.