The size of the Roman army varied with necessity, and "the height of its reign" is a bit vague, so the following is just a snapshot from the time when the population of the empire was at its height.

David Potter gives a population of "somewhere in the vicinity of 64 million people by the time of Marcus Aurelius" (second half of the second century). (p. 17)

Peter Kehne gives an army size just prior to the reign of Septimus Severus (late second century) of "480,000 soldiers, sailors, and others". This includes legionaries (30 legions), auxiliaries, 12,000 soldiers stationed in Italy, 30,000 sailors and marines.

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Sources:

• D. Potter, The Roman Empire at Bay AD 180-395 (London 2004)
• P. Kehne, War- and Peacetime Logistics: Supplying Imperial Armies in East and West, in: P. Erdkamp, A Companion to the Roman Army, (Oxford 2007), p. 323-338