The Tower of Babel.
Well, you probably won't buy that, but I'll give it my best shot (I'm just finishing up an M.S. in Architectural History). And I'll try to throw in some random historical facts as I go along. Sorry, it's in no particular order.
We'll say that the earliest elevator as we know it and in general use came along around the middle of the ninteenth century.
Source
Anything of real height built after that -- at least things that we'd call buildings, not merely iron communication towers or something -- incorporated an elevator.
Almost all of the multiple story buildings in Chicago and New York in the 1880s and later used elevators -- in fact, the presence of an elevator is generally considered one of the defining characteristics of a skyscraper. This applies to both masonry-wall and steel-frame structures.
The Eiffel Tower was the tallest structure for most of the 19th century, however, the design incorporated an elevator.
My bet, after thinking over this for a bit, is that the winning structure is the Great Pyramid at Cheops -- height ~ 480 ft. This crushes the height of other particularly large structures without elevators -- and here I'm thinking about the Cathedral at Beauvais--the highest of all of the Medieval Cathedrals, which Google puts at ~158 ft. Although none is as high, second place finishes might be had by various pyramid/ziggurat structures in South America or Messopotamia.
Now granted, this doesn't answer your question if you're talking about "what's the heighest accessible space in a building without an elevator." Then, it would most likely be one of the Cathedrals of the Middle Ages, or perhaps a tower built in the 19th century. One quasi-inhabitable space was done as a Cathedral/folly/house in England, but it collapsed and was never rebuilt. If you're really interested, I can try to look up the name of it for you.