The roman army was comprised of 33 legions in 200AD. In 30BC, Augustus formed his legions into nine "cohorts" of 480 men, one "double strength" cohort of 800 men (the first cohort) and a cavalry squadron of 128. Therefore there were around 5,248 men in a theoretical legion, including various officers. On paper there were 173,184 men in the legions in 200AD, although legions were frequently undermanned thanks to disease, battle, retirement and a number of other factors. In addition, under Septimus Serverus, in 200AD, there were around 400 units of 500 auxiliaries, making an additional 250,000 men.

Therefore, as an overestimate (due to understrength units), there were 423,184 men in the roman army at its peak. This figure does not include other forces such as mercenaries and naval forces.

References: Legions of Rome, Stephen Dando-Collins, 2010, ISBN 9781849162302.

That's rather difficult to say. First of all, we need to establish what we mean by "peak." It seems pedantic, but are we talking about the period of greatest size, maximum power, largest territory occupied, most successful wars, etc? All of these would produce a different answer. I'm going to assume you mean largest size, which I think is the most basic assumption (correct me if I'm wrong). However, you're going to see that even then the answer is not as straightforward as simply spitting out a number of soldiers, either mentioned from a text or calculated according to a known or estimated number of legions. I'm also going to limit the scope of our investigation to the period after the Punic Wars, because while there are some extremely large and impressive battles being fought during the period before, military organization and procurement were such that it makes them largely irrelevant. Still, you'll see how difficult this is to say.

So the first thing to understand about the Roman army is that there was no Roman army until the Augustan army. During the Republic military forces were legally only allowed to be controlled by magistrates holding imperium. Until Rome controlled provinces this was pretty much limited to consuls and praetors, although extraordinary magistracies occasionally were enacted even at this early period. Now even if we narrow our focus to the period following the abolition of property requirements, which allowed Roman armies to stand in the field more or less indefinitely, we still have a problem. Even under this system, and with the addition of propraetorian and proconsular magistracies holding imperium as provincial governors, most armies were still not standing, nor were they state-controlled. A magistrate, once granted imperium and given his legal access to military tribunes, raised and organized his own armies (paying for them theoretically with the treasury grants given to him, as well as whatever he could exact from the cities of his province, but in reality large armies were so expensive that private funds had to be used to bankroll them almost always). These armies were then his to do with as he pleased, provided that he didn't break the law, with only nominal input from the senate (which remember was purely an advisory body and had no legal power to meddle with the affairs of magistrates). Even in the case of a magistrate overstepping his legal bounds, such as Gabinius' expedition into Egypt during his governorship of Syria (or, arguably, most of the events of Caesar's proconsulship--all of them if you accept Cato's accusation that his magistracy was illegal in the first place), there was nothing the senate could do immediately (except maybe demand the magistrate's recall, which a magistrate could delay or even refuse--although doing so was sure to destroy his career) and even after the end of his magistracy there was no guarantee that prosecution would succeed. A magistrate was expected to raise the troops he needed, and then disband them (as both Piso and Cicero did), but that didn't always happen. The "Roman army" of the Republic is a collection of private magisterial armies operating independently, funded independently, and only nominally doing the bidding of the state as a whole. They aren't even necessarily magisterial, or legal, for that matter--Pompey was supposed to disband his army after his extraordinary magistracy in Asia Minor ended, but refused to do so, and Octavian built a gigantic personal army by persuading Caesarian legions to defect from Antony despite having no official magistracy, which made his army illegal until Cicero granted him an extraordinary magistracy of consular powers to hunt down and destroy Antony. Because of this the total military power of the Republic was gigantic, with each individual magistrate being able to field armies as large as they could afford. If we were to add up all of these troops then we come up with some staggering figures--at Pharsalus alone Pompey and Caesar fielded together more than 60,000 men, and at Philippi there were literally hundreds of thousands of men on each side. But should we do so? These troops are not unified, they're frequently not even supposed to exist, and they aren't acting together as some sort of state strategy--they aren't even fighting on the same side sometimes! That makes this hard, because the late Republic is without a doubt the period during which the most Roman soldiers were in the field at one time--Octavian had more than 60 legions after Actium, and the combined legions during several years of the various prior civil wars undoubtedly exceeded even that absurdly large figure. But I would argue that this shouldn't count, and that each private army should be considered quite distinctly from the others, as they are very distinct entities.

Ok, now on to the Augustan army and the armies of the Principate, which are more relevant to your question. When Octavian won the War of Actium he found himself holding a bag containing every surviving legion in the world, an enormous number of soldiers. Octavian realized perfectly well how easy it was to exploit the system of Republican magistracies to build gigantic bases of power and huge armies, having exploited it himself both legally and illegally. His reduction of the number of legions allowed in the field at any time (barring permission from himself for additional troops) was mainly to get these soldiers out of soldiering (and therefore keeping them from continuing to be a menace to society) and to keep ambitious governors from trying to gain too much power and do what, well, he and Caesar had done. So we see that Roman military forces become reduced to twenty-some-odd legions (the exact number varies over time and according to the military commitments of the empire, but the changes are really purely academic). More important is the fact that under Augustus the army answers not to the various magistrates holding imperium, but to Augustus personally, which continues through the succeeding emperors. Military commanders are still magistrates, but they are imperially-appointed magistrates who have to answer to Augustus, who's holding all the cards. It's at this point that I think we can begin to speak of a "Roman army," because from here up until the 3rd Century (barring 69 AD, which should be discounted for the same reasons the armies of the civil wars should) the Roman army is firmly in the control of a single authority.

To be continued...

Follow up question: How large would a single battalion (or grouping, rather) be at this time?

Depends on what you mean by 'peak' really. I'll try and give you an overview for the Empire period.

When Augustus came to power by finally defeating Mark Anthony he joined all of the previously competing forces together and cut the overall army size in half, down to about 250,000 men half of whom were in the legions (citizen heavy infantry) the other half in the auxiliaries (non-citizens who filled a number of combat roles). This number gradually grew and by 115 AD during the reign of Trajan the armed forces had increased in size to about 400,000. It was during this period (Pax Romana) that the Empire and army are usually considered to have been at their 'peak'.

The army was devastated by plague, invasions and especially civil wars during the crisis of the third century but was rebuilt by Diocletian, who also greatly changed organisation, equipment and pretty much everything else. While the later army was traditionally considered to be 'barbarised' and greatly weaker than it's predecessor most modern historians believe (and I agree) that it was still a formidable force. Estimates about the size of the army in this period differ between a low count of 400,000 and a high count of 600,000.

Roman organisation of soldiers into standardised groupings was often much different on paper than in reality, with illness and leave generally leaving units below strength. Anyway the smallest grouping during the Principate was a contubernium (tent party) of 8 men. 10 of these formed a century (it's possible that 20 non-combatants would be attached to the 80 soldiers to make the numbers up to a hundred, thus making it an actual century). In turn 6 centuries formed a cohort of 480 active combatants. Finally 9 regular cohorts and a single elite First Cohort (of 800 men) made up the largest organised unit; the Legion which became synonymous with Rome.

If you're interested in learning more about the Roman military I'd recommend Adrian Goldsworthy's 'The Complete Roman Army'.