Unsepttrium
Although Richard Feynman noted that a simplistic interpretation of the relativistic Dirac equation runs into problems with electron orbitals at Z > 1/α = 137, suggesting that neutral atoms cannot exist beyond untriseptium, and that a periodic table of elements based on electron orbitals breaks down at this point, a more thorough analysis calculates the limit to be Z ≈ 173, meaning that neutral atoms most likely cannot exist with atomic number greater than 173.  This makes unsepttrium theoretically the heaviest neutral element possible.