Technically, five.
Our Thief Rogue 17 (Ace) has three Battlemaster friends (Bart, Carol, and Dave). They roll initiative. Bart is first followed by Ace, then Carol, then Ace again, then Dave.
Bart uses Commander's Strike to allow Ace to sneak attack (1) with his Reaction.
Ace starts his turn, so his Reaction is ready again. Ace sneak attacks with an Attack action (2).
Carol uses Commander's Strike to allow Ace to sneak attack (3) with his Reaction.
Ace starts his turn, so his Reaction is ready again. Ace sneak attacks with an Attack action (4).
Dave uses Commander's Strike to allow Ace to sneak attack (5) with his Reaction.
This uses (abuses, really) the Thief's ability to take two entire turns during a single round, and also "steals" a reaction from the next round due to the wibbly-wobbly mechanics of how action recharge doesn't quite match up with rounds.
Mechanical abuse that doesn't quite match up with the spirit of the question would be to use a scroll of Shapechange to turn into a Marilith (credit to Miniman), which can take a reaction every turn of combat - thereby allowing one sneak attack per active combatant.
If multiclassing and having a very weak sneak attack is allowed, the possibilities are Infinite.
A Fighter (Cavalier) 18 / Rogue 2 can make an opportunity attack on every turn without using his reaction, and could feasibly sneak attack an infinite number of creatures (Arcane Gate or other mass translocation spells can provide ample space for creatures to start their turns close enough to move in/out and provoke) who choose to provoke opportunity attacks from him.