Supposing each regeneration is say up to around 1000 years or more and each is treated as a "new life" almost couldn't he be thousands of years old by now?
Except each regeneration doesn't last 1000 years.
Okay, we don't know how old the First Doctor is at the start, but he is clearly old, old enough that he eventually regenerates of natural causes.
The Second Doctor has human companions with him his entire time, and since they never age we can infer that no more than the three years that went by in the real world went by in the show.
Not only does the Third Doctor have human companions with him constantly who do not age, he spends most of his time on Earth without time travel ability for a decent percentage of it, so again probably not much more time went by in the show than in the real world.
With the Fourth Doctor there is some wiggle room, given he briefly spent some time alone and some time with a Time Lady for a companion, meaning decades could have gone by with them not showing age. Still, I'd be surprised if he even made it to the century mark in this form.
The Fifth Doctor and Sixth Doctors had humans with them the whole time who don't age and don't have any time alone, limiting how much went by in their respective lives.
The Seventh Doctor clearly aged a bit when he regenerated and given there were no companions with him when he regenerated could very well have lasted decades.
The Eighth likewise aged by Time Lord standards at least several decades worth between the TV movie and Night of the Doctor. Plus given how much we know went on during his lifetime (the audios are canon, given they are mentioned in Night, plus the Time War started) he could be one of the longer lived Doctors.
The War Doctor definitely aged a lot in his lifetime, and was the first Doctor since the First to have regenerated due to natural causes. So definitely past a century, possibly even a few centuries.
The Ninth, we'll start with RTD's assertion that Rose isn't his post-regeneration story. So, there's likely a few decades between regeneration and Rose, but the period from Rose to Parting of the Wars covers less than a year for him.
The Tenth we get a reasonably accurate measurement of how long he was around, since the RTD era stayed very consistent with the Doctor's age. Starting with when the Ninth said he was 900, the Tenth consistently added another year to that each season, ultimately saying he was 906 just prior to regenerating.
The Eleventh things get, tricky. In the Weeping Angel story he says he's 907. By The Impossible Astronaut he's 909, and the future version of himself, which is him at the end of the season is 1100. By season 7 he's 1200, which is where he stays up until Trenzalore. Since he spends 900 years on Trenzalore, he's 2100 by the time he regenerates.
So far the Twelfth Doctor hasn't really elaborated on his age other than to say he's over two thousand years old.