Based on the wording of the card, it would seem that you still only get one extra turn.

See this answer in a video!
Little Answers

A: Correct. You can’t save up any extra Another Turns. The maximum number of turns you can take in a row using this card is two. Most, but, unfortunately, not all, iterations or versions of this card have that language on them. Keeping things consistent across so many Fluxx decks is surprisingly difficult.

For those wondering how this card could be played more than once, it’s quite possible that one could play Take Another Turn, and then Let’s Do That Again (either during their first turn, or their “Another” turn)… but you can’t take another “Another Turn,” so don’t pick that card to Do Again.

Shadow Out Of Time in Cthulhu Fluxx functions the same way.