At what level in 1e/2e did people fight balors and liches?
From everything I've read and IME, most people did this at levels 9-13.
So what non-unique opponents were you supposed to fight after level 13?
There was virtually nothing in the game which could be used unmodified to challenge a typical party beyond the 10th level. The game just wasn't designed for that. At 10th level and above, there was an assumption that characters would spend more time 'solo' adventuring (along with perhaps henchmen) or that they'd retire.
The monster manual(s) classified monsters from I to X. First level characters were assumed to mostly encounter type I monsters, 2nd level type II's, 3rd level type III's and so forth. Above 10th level, there wasn't much that could be a challenge. Typically challenges were built from the following at those levels:
1) NPC's - Opposing high level characters could be made as powerful as you wanted. Drow seemed to get this treatment all too often.
2) An assortment of high HD or abusable monsters, many of which can be seen being abused in the published module 'Axe of the Dwarven Lords'. (The 1st edition versions were often more abusable). One trick was to look for things that ignored AC or had attacks that didn't offer saves or had attacks for which level wasn't a defense. The 1e Korred was a particularly abusable for example because its laughter attack actually worked best against HIGH charisma. Off the top of my head that list would be things like: max-headed hydras of all classes, larger giant races, huge ancient dragons, beholders, thessalhydra, largest size remorhaz, grey and death slaad, pit fiends, shambling mound, arcanodaemon, vampires with class levels, liches, golems, black pudding, crimson death, largest size barghest, rakshasa lord. Mephits could be surprisingly useful as mooks as many had attacks that did damage even if the save was passed or could use autohit spells like magic missile. And they flied, etc.
3) Take advantage of typos: The best example of this would be 82 HD Jann lords.
4) Make stuff up - Gygax seemed to go with this the most when challenging characters above 10th level, and many DMs followed suit. Doubling or tripling the HD of ordinary monsters made legendary versions of the same and had some justification from the text (Talos, for examples). New monsters with extraordinary abilities could always be made up - demiliches and giant ape gods for example.