How far off the baseline can a runner be if there is no play on him?

No, he cannot. The only time he can leave the 3 foot baseline is to avoid colliding with a fielder who is attempting to field the ball.

This rule is also supposed to include runners who purposely slide out of the baseline to break up a double-play, but sometimes Umpires forget how short of a distance 3 feet is.

Here is the official rule:

Rule 7.08 (1)

A runner is out when he runs more than three feet away from his base path to avoid being tagged unless his action is to avoid interference with a fielder fielding a batted ball. A runner's base path is established when the tag attempt occurs and is a straight line from the runner to the base he is attempting to reach safely.

This rule also helps to explain why it is okay for a runner rounding a base to swing more than 3 feet out of the baseline as they turn, because there is no tag attempt being made at the time they are out of the 3 foot boundary.

If, for any reason, the batter is already out of the baseline before a tag attempt is made, they are not allowed to Dodge the tag attempt by moving back to the original baseline. Where they are currently at when a tag attempt begins is where the new baseline is established, and they cannot run 3 feet out of where they currently are. They must attempt a 3 foot wide straight line from their current position to the base they are trying to reach.