With typical tile-based maps such as OSM's default Mapnik style, there is a cache to store previously rendered tiles, but it contains just a small subset of all possible tiles. Other tiles are calculated on-the-fly if a client tries to access them. Only a small percentage of the tiles are actually ever requested as most of them are likely not particularly interesting to users.
The OpenStreetMap wiki's page on Tile Disk Usage has some numbers with a breakdown by zoom level. According to this source, you are looking at roughly 50 TB of data if you want to store all tiles for a global map with 18 zoom levels. Other estimates range into the hundreds of TB, though.