Do you have to serve two jail sentences? For example, you go to jail the first time and use your "Get out of jail free" card, and then go right back to jail for another sentence without rolling the dice.
You go to jail directly on rolling the third double, so there’s no opportunity to land on “go to jail”.
From the Official Monopoly Rules (Hasbro)
If you throw doubles, you move your token as usual, the sum of the two dice, and are subject to any privileges or penalties pertaining to the space on which you land. Retaining the dice, throw again and move your token as before. If you throw doubles three times in succession, move your token immediately to the space marked "In Jail" (see JAIL).
You're thinking about this too much in terms of real-life prison sentences, and too little as game mechanics. "Jail" is simply a location on the board, associated with a few specific rules, but none of those rules suggest that there would be "Jail Time" stacking up. Instead, you are either in Jail, or not.
Furthermore, upon rolling doubles a third time, you are directly placed in Jail without executing your movement anyway:
When doubles are rolled, the player resolves the roll as normal (including purchase, renting or passing "GO"), but rolls the dice again for another turn (you have to keep going). The player moves forward as directed by the dice, and if this is also doubles, rolls again. If the third dice roll is doubles, the player cannot buy property, and is instead moved directly to jail. [source]