How does a riding mower recharge the battery?

Most don't, having a magneto to power the spark plug. Larger garden tractors can have a generator/starter that functions both as a generator when running and a starter for starting. The output is controlled by a voltage regulator and charges the battery. There is also usually an ammeter or other charge indicator with this type of system.

I really hate to disagree with another expert. But Scleros is absolutely incorrect. The starter only acts as a device to turn the engine until it starts. Then the starter just sits there and enjoys the ride. The starters only function is to start the engine.
Now for the answer: the engine has a stator underneath the flywheel that is made of windings of wire with insulators in between, the stator is mounted to the engine's body and is stationary. The flywheel has magnets spaced around the interior surface of the flywheel which of course moves around as the engine turns. By the breaks in the stator (insulators) and the spacing of the magnets on the flywheel as the two pass one another they produce the energy to recharge the battery and also to power accessories.
Start the engine, turn on the head lights and lower the rpms, notice the lights are dim at idle. Then rev the engine to top rpms and notice the lights are bright. That is because of what I just explained about the magnets and the stator. The quicker the engine turns the more energy that is made by the stator.

Most don't, having a magneto to power the spark plug. Larger garden tractors can have a generator/starter that functions both as a generator when running and a starter for starting. The output is controlled by a voltage regulator and charges the battery. There is also usually an ammeter or other charge indicator with this type of system.

The only correct part about this is that the magneto (armature magneto) does indeed provide the spark to the spark plug.

For a generator/starter system, have a look at an older Cub Cadet, or see Starter-Generator (http://www.simpletractors.com/service/electrics/starter-generator.htm). I may have left the stator systems on mid size riders out of my response, but I'm not incorrect. There have been a wide range of electrical systems over the years depending on the size of the "riding mower". I'll also concede that my 'Most" would have more accurately been a "Some". I post what I know and am most familiar with. You're free to do the same and add additional info, however in the future before disagreeing, be d@#$ sure you know all you think you know.

Yes it was overall a no good, very bad, hasty post based on the low and high ends that I had sitting in the garage and not representative of the larger rider market as a whole. I also failed to mention electrics that you plug in to recharge.