In LBB-only OD&D, it's stated that "There is no theoretical limit to how high a character may progress." However, the game's tables stop at 10th level for clerics and fighting men and go to 16th level for magic-users. Guidelines are provided for "levels above those listed," but those guidelines only cover hit dice, fighting capability, and spells. The number of experience needed for those additional levels isn't stated, assumption being, I suppose that one can intuit an appropriate rate of advance based on the existing tables.
Supplement I includes expanded tables for both clerics (to 20th level) and magic-users (to 22nd level), in part because of the introduction of new levels of spells for both classes. The thief class is also introduced, but its level progression table ends at 14th level. There is no new chart for fighting men. Interestingly, the next two OD&D supplements, Blackmoor and Eldritch Wizardry, introduced new character classes that had explicit level limits: assassins (14th level), druids (13th level), and monks (16th level).
I bring all this up because I've often wondered why it is that we so often hear D&D players complain that the game "breaks down at high levels." That statement may in fact be true but, in all my years of playing D&D, it was rare that a character ever legitimately achieved a level higher than 12 and even then such an accomplishment was rare. The highest level character ever played in any of my pre-WotC D&D campaigns was 16th and he was a noteworthy exception. In my Dwimmermount campaign, after 2+ years of play, the highest level PC was nearly 8th level and, owing to death and attrition, most PCs were a couple of levels lower than that.
One wonders, then, where all these high-level characters have come from. Have I been doing something wrong all these years? More to the point, was D&D written with the expectation that PCs would reach much higher than 10th-12th level? My own feeling is that it wasn't, even if the game leaves open the possibility of such a thing -- but then the game also leaves open the possibility of dragon and balrog PCs, too, so I'm not sure that says much.