Question: In 5e, a player who is grappling has their movement speed halved. However, a player with two hands can grapple *two* opponents.

As the wording seems to indicate that the movement speed penalty applies due to the grappling status, does that mean that a player grappling two opponents is doubly penalized?

If so, how would it apply: is your speed reduced to 0 ([1 - 0.5] - 0.5), or a quarter of your regular speed ([1 / 2] / 2)?

Answer: 
It's reduced to a quarter of your regular speed
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> 
> ***Moving a Grappled Creature.*** When you move, you can drag or carry the grappled creature with you, but your speed is halved, unless the creature is two or more sizes smaller than you.
> 
> 
> 

So when you want to start dragging one creature, that will halve your speed. If your speed was 40 its now 20. In addition to dragging the first creature you want to drag a second creature, which again applies the halving, so your current speed of 20 is now 10.
It Does Not Stack
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It is very important to know exactly what the rules say here.

PHB, pg. 195

> 
> **Moving a Grappled Creature.** When you move, you can drag or carry the grappled creature with you, but *your speed* is halved, unless the
>  creature is two or more sizes smaller than you.
> 
> 
> 

Note here that it says "your speed" and not "your remaining speed" -- your speed is always constant.

So if you are a human with a speed of 30 ft, *your speed* is halved to 15 feet. If you grapple a second creature, *your speed* is halved again to 15 feet -- but it doesn't matter at that point, since it was already halved the first time to the exact same value.

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A related question which explains how the movement speed penalty works: [How Do Grapplers Stand If Prone?](https://rpg.stackexchange.com/questions/79392/how-do-grapplers-stand-if-prone)