There's no limit built into the Bowling lane itself; you could in theory line up a hundred (or more) people, letting each one roll one ball and then leave.
However, most modern automatic scoring systems will only allow you to enter up to six names per lane. And that's is also a good number in terms of time: six people will take on average about an hour to finish one full game.
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That would all depend on how many lanes the bowling alley had. Some have 40. Some have 100. There is one in South Tampa that has 4. Usually you can comfortably fit 6 bowlers on a lane if time constraints are not important. When I worked in an alley, we tried to keep open bowlers to four to a lane.
There isn't a specific number of bowling balls needed in a bowling center. Technically, the center could require those who bowl to bring their own.
Generally the center will try to stock a range of balls of different weights and drill patterns. Depending on the investment, some centers will try to have 3-4 balls per lane at the minimum.
It all depends on the bowling alley. There are really private alleys with just 2 lanes or a bowling stadium with 73+ lanes.