It depends on the edition.
In BECMI, they start at just barely above the level of the common man, but gain power quite rapidly. Non-spellcasters plateau around 10th level; spellcasters continue to gain power (via new spell levels).
In 1st Edition, characters are a clear step more powerful than their BECMI counterparts at all levels. The pattern (and rate) of power gain is otherwise the same.
2nd Edition was a real rarity: an edition that actually lowered the power level. 2nd Edition PCs stand somewhere between BECMI and 1st Edition in terms of power. The pattern (and rate) of power gain is the same.
In 3e, the PCs are a step more powerful than in 1st Edition. 3e also removed the plateau for non-spellcasters - all characters gain power at all points in the level range. The rate of power gain is otherwise similar to previous editions.
(Note: the range of specialised options available in 3e means that 1st level characters can be as good as the very best of real-life humanity. So, characters are effectively superhuman pretty much as soon as they reach 2nd level.)
In 4e, PCs start a good bit more powerful than in 3e, but they gain further power at a reduced rate.
(Note 2: it is now very common for 4e PCs to start with a '20' in their primary stat. Because of the way the skill system is set up, this effectively means that every Fighter is the World's Strongest Man, every Wizard is an Einstein, every Bard is an Elvis, and every Warlord is a Julius Caesar. And all of this at 1st level, before they've actually done anything!)
In 5e, I would like to see this recalibrated a bit, so that PCs start "a cut above" the masses, but not superhumanly so. They should then gain power at a moderate rate, becoming "larger than life, but not absurdly so" around the end of the Heroic tier, and then becoming "legendary heroes, who do six impossible things before breakfast" around the end of the Paragon tier.
In BECMI, Aragorn is probably around 9th level. In 1st Edition, he's probably about the same (everything is just bigger). In 2nd Edition, he's probably gone up to 10th - 12th level. In 3e, Aragorn has dropped to 5th - 7th level. In 4e, Aragorn may be as low as 1st level, seriously. (In 5e, I would like to see Aragorn be a mid-Paragon character.)