If you check the section "Extensions of the standard dictionary numbers" on the page you linked, there's a system for naming arbitrarily large numbers as -illions. The prime here is approx. 8.82 * 10^41024319, and 10^41024319 is the 13674772nd -illion (short scale), so it would begin eight tredecilliquattuorseptuagintasescentilliduoseptuagintaseptingentillion.

The whole point of the extensible system is that you can, in fact, extend it as much as you need.

... I just threw this into my zillion-ifier program and ... huh, it took 12 hours even with the GMP version, though I'll admit I didn't really optimize it ... the output is about a gigabyte of text

  eight tredecilliquattuorseptuagintasescentilliduoseptuagintaseptingentillion,
  eight hundred sixteen tredecilliquattuorseptuagintasescentilliunseptuagintaseptingentillion,
  nine hundred forty three tredecilliquattuorseptuagintasescentilliseptuagintaseptingentillion,
  two hundred seventy five tredecilliquattuorseptuagintasescentillinovensexagintaseptingentillion,
  thirty eight tredecilliquattuorseptuagintasescentillioctosexagintaseptingentillion,
  ...