India, who resumed the day on 391-4, advanced to 759-7 before declaring their innings. England were 12 for no loss at the time of stumps, and trail India by 270 runs with only the final day of play left.
Nair -- who recorded his maiden Test century during Monday's first session -- showed great composure throughout the day as he gradually built up his innings.
Nair speeded up his innings in the day's last session, and smashed the England bowlers all over the park in the final few overs of the day to complete his triple ton.
The Karnataka batsman is the second Indian batsman after Virender Sehwag to score a triple century, with the latter having two to his name.
Earlier in the day, India lost Murali Vijay to a Liam Dawson delivery in the day's first session, but Ravichandran Ashwin ably supported Nair to help India cross 600 runs.
England skipper Alastair Cook's tactic of rotating his bowlers on a flat pitch at Chennai's MA Chidambaram Stadium restricted India to just 72 runs in the first session but the Indian batsmen stepped on the gas after lunch.
The hosts added 119 runs in the afternoon session as England bowled 30 overs without getting a single wicket.
Ashwin was dismissed for 67, but Jadeja scored 51 as Karun Nair closed in on his triple ton.
India finally declared their innings after Nair completed his triple hundred.
The hosts have already secured the series with a 3-0 lead, and will look to hand England a 4-0 humiliation when play resumes in the final day of the series on Tuesday.
(With inputs from AFP)