The Romans did build (relatively) high rise apartment complexes, called insulae. To quote Vitruvius (De Architectura, II, 8, 17) :

"Given the importance of the City and the extreme population density, it is necessary tu multiply housing. As single-story housing could not suffice to house such a massive population in the city, it was necessary, in that situation, to build in higher floors."

It has been estimated that during Constantine's time, Rome would have had around 26 000 such insulae^1. The remains of such buildings can still be seen in Rome (for example in the Capitole close to Santa Maria in Aracoeli) or in Ostia^2.

Regarding the floor count, the remains of insulae found in Ostia are typically 3 stories high, corresponding to a typical composition of the building with tabernae (small shops) on the ground floor and two upper floors with housing^2. Yet Strabo tells us that there was a ban on building higher than 70 roman feet above the street level (around 21m), due to concerns about structural stability and fire risks :

« Caesar August took care of preventing incidents, first by recruiting a militia corps of firefighters among freed slaves and reducing collapses by reducing the height of new constructions, forbidding any new building higher than 70 feet above street level» (Strabo, V, 3, 7).

Later, Aurelius Victor mentions that in the time of Trajan, it was forbidden to build higher than 60 feet (Aurelius Victor, Epitome de Caesaribus, XIII, 12). 10 Roman feet would be around the height of a floor (3.3-3.5m), meaning that it was forbidden to build higher than 7 floors, then 6.

Interestingly, this is roughly the maximal number of floors you would find in a Haussmannian building (second half of the XIXth century in Paris) : the zoning regulation of 1884 gives a maximal height of 20m above the major avenues (and less above smaller streets), and only the first 4 or 5 floors would be inhabited by bourgeois families, with the last floors made up of chambres de bonnes, that is rooms for domestic workers. There is a simple reason for this : neither Haussmann nor the Romans had elevators. If you need to take by foot your heating fuel (wood in the roman era, coal in XIXth century France), your water, and yourself to your housing, climbing more than 6 or 7 stairs becomes quickly impossible. Incidentally, the typical khrushchevka in the former Soviet Union (housing unit building from the 50s) is 3 stories high because it was the legal maximal height admissible without adding elevators, which were a luxury the soviet economy could not have scaled up to cope with the extremely high rate of housing construction in that era [surely a redditor more familiar with Soviet history would be able to provide detail on this].

To conclude, this height was both a safety regulation and a practical necessity : in an era without elevators, living in housing higher than 6 or 7 stories becomes quickly difficult

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^1 : Habitation, Lexique d'Histoire et de civilisation romaines, Jean Luc Lamboley

^2 : Pierre Gros, L'Architecture romaine du début du iiie siècle av. J.-C. à la fin du Haut Empire, 2. Maisons, palais, villas et tombeaux, Paris, Picard, 2001