Extended game
For the long defunct spell, see Extension.
The extended game (15 rune game, or simply extended) comprises of the dungeon branches outside the scope of an ordinary ascension.
To win the game, you only need 3 of the 15 possible runes of Zot. Most players go to the two easier Lair branches, then one (or more) of Abyss, Slime, or Vaults. The other rune branches are considered the "post game", i.e., the extended game. They are designed to be difficult even for XL 27 characters, and most likely harder than the other options. Grabbing all 15 runes can be done for score, tournament points, bragging rights, etc.
Branches
Pan:
Hell:
Tomb:
Occasionally, the Abyss and Slime Pits are also called part of extended. Meanwhile, the Crypt and Vestibule of Hell are reasonable to do outside of it. This can change depending on character build (Slime is definitely not "extended" for Jiyva worshippers).
Strategy
The extended branches are meant to be challenging (or at least risky) even after clearing the Realm of Zot. For maximum safety, complete Zot (at least Zot:1-4) and any leftover rune branches first.
As you progress through the extended game, you'll find yourself with a surplus of XP. Therefore, your exact skills matter much less, so you can take your character to fairly wild directions. This is also a good time to change your god, if at all - most characters are able to handle divine retribution from most gods at this point. Abyss and Crypt are good places to shrug off the godly wrath.
Most of extended is filled with two threats: torment and damnation. Torment directly halves your current health, and resisting it is difficult if you're not already undead. Damnation, mostly found in Pan and Hell, ignores AC, EV, and SH; players can never gain a resistance to it.
For more specialized information, see each branches' specific page.