Answer: 1-2mm when new. Additional play over the course of use can vary.
Dropper seatposts rely on a small gap between the brass guide keys and keyway slot to move freely and stay properly lubricated when the post is actuated. Rotational or side-to-side play in the dropper stanchion (and seat) are a result of this gap. This gap is also important to allow contaminants (dirt and grime) to be purged (as opposed to binding and damaging) from the keyway. Dropper posts with a tighter key vs keyway fit and subsequently less rotational play often are more service intensive and problematic.
A new post will have 1-2mm when measured at the nose of the saddle and this is considered normal. Over the course of the first few rides, this play might increase incrementally as the system breaks in.
A multitude of factors can impact the speed at which the brass guide keys wear and play increases as the post gets used. Seat tube angle, riding conditions, rider weight, and how regularly the post is getting serviced will all impact the rate at which brass guide keys wear and stanchion rotational play increases.