That all depends on whether you mean a Regular army, or if you want to also include the various state militias and the like.

According to the text of Historical register and dictionary of the United States Army : from its organization, September 29, 1789, to March 2, 1903 , published in 1903, the United States in 1790 had a single regiment of infantry and a single battalion of artillery. This army of 57 officers and 1,273 men, totaling 1,330 men of all ranks, was the entirety of the United Stats Army authorized in April, 1790 (page 560).

A Legionary system was set up in 1792, to consist of four Sub-Legions. These forces summed to 258 officers and 5,414 enlisted men. Each Legion would consist of 57 officers and 1,346 enlisted men, quadrupling the army of 1790 (p 562). Over the years that followed, the Legion was reformed and downsized by Congress in 1796.

1798 was a massive year of expansion. The Army went from a strength of 4,448 officers and men in the approved size in the authorization of April 1798 (p 564) to an authorized size of 15,204 men by July. This massively increased army consisted of a Corps of Artillery and Engineers, a Regiment of Light Dragoons, and sixteen regiments of infantry (p 566).

1799 brought another reduction of forces, and by May of 1800, the United States Army was down to 4,754 officers and men (p 568).

Now, to answer the rest of your question about a theoretical maximum military size for the United States.

The US Census of 1800 gives us much more granular detail compared to the 1790 Census, and thus we'll be using those numbers, as we look at the 825,744 white free men between the ages of 16 and 45. We can extrapolate a back-of-the-napkin estimate of 275,000 men under arms - that is, putting no more than approximately 1/3 of your eligible total of military-aged men to the colors.

Now, how does this compare to other countries of the time? Well, discounting slaves (who made up 893,605 of the nation's 5,305,982 total population), we come to 4,412,377 people within the United States in 1800. You're looking at approximately 1/6th of France (26 million): who mobilized 1.5 million men at their peak.

As a percentage, that is approximately 5.76% of their total population at their greatest mobilization level. If we assume that extraordinary circumstances would permit, in 1799, for the United States to reach that same threshold, it would equate to the fielding of 254,560 men. This equates to approximately 2/3 of all males aged 16-26, or 30.82% of the eligible adult male population 16-45. It is also, rather conveniently, beyond this value that you start to put a serious strain on the home front, and thus short of a total national mobilization for war the likes of which was never seen prior to World War One, really, it also serves as the maximal number of men the United States could put into the field.

Further, the vast majority of those that would be called to arms would be, by and large, at least nominally trained, because the Militia Act of 1792 stated that "each and every free able-bodied white male citizen of the respective States, resident therein, who is or shall be of age of eighteen years, and under the age of forty-five years (except as is herein after excepted) shall severally and respectively be enrolled in the militia".

Now, in peacetime settings, a maximum of 1-2% of a nation's military ever serves under arms unless there is mandatory peacetime conscription. The United States in this case, there's a maximal peacetime military of about 90,000 men at 2% military participation, or 88,000 men. The 1798 army stood at 15,000 men, or one quarter of the maximal by that value. But it was basically the greatest practical limit of a peacetime army out of political considerations than any manpower matters. Further, many of those remaining 73,000 able-bodied men that would have taken up service would have more likely done so in State or local militias.

So, in a situation that suitably threatened the United States with annihilation, much as France was threatened with her own destruction during the Wars of Revolution: the United States could levy approximately 250,000 men for itself. At a practical level however, the 15,000 men of 1798 was the greatest level that the United States could reasonably rally in peacetime as a force answerable directly and solely to the federal government.