How many times can you split the 3.5mm audio line cable signal from a music player
TL;DR You can split to any number of amplifiers you care to, but only a limited number of headphones. Try it on your particular player and see how it copes, as there will be differences.

A music player output tends to be able to drive a pair of 32 ohm headphones to reasonable levels.

This means that the output impedance of the player is low compared to 32 ohms, often a factor of 10 lower. The output impedance is important as it governs how much voltage drop there is on the output when it's loaded. 

The input impedance of an amplifier is usually in the 10k to 100k region. There is essentially no voltage drop, regardless of the number of amplifiers you split the signal to (well, up to hundreds of them).

With a factor of 10 difference in impedance, there could be a 10% drop in output level when headphones are connected, compared to when they're not connected. 

Connect a second pair of headphones, and there's a further drop. Compared to the variation in volume of music, and the variation possible due to the volume control, you would still be hard pushed to notice the difference of adding a second pair.

If a small reduction in volume was the only issue, then you may think there would be little problem in splitting the signal to more headphones. However, there are several issues.

Music players tend to have a limited current output. As more headphones in parallel draw more current, that will eventually overload the amplifier, causing clipping distortion. This will occur at a lower sound level, the more sets of headphones are driven.

Music players might go unstable into lower impedance loads, and you would get poor sound quality regardless of volume.

If it's smart enough, a music player might shut down into a too low impedance load, and then you'd hear nothing.