I was playing a session where a player says that a character can never be brought below half speed and the DM agreed. I've not read anything that suggests that speed cannot be lowered beyond half speed. It seems reasonable to me that if a person's movement is halved due to rough terrain could have his or her movement reduced by half again by being effected by a slow spell, meaning that the character is effectively down to 25% movement. Thank you.

You can definitely be slowed beyond half. You just can't stack difficult terrain with more difficult terrain.

There are two arguments here -

1. can player speed ever be reduced below half?
2. can a player be slowed by different things at the same time?

1) is easy. Aside from obvious things that reduce speed to 0, there's plant growth.

2) I think is what you're really arguing about, and since the rules don't seem to address it directly, you need to determine the default assumption. Because 1 is yes and there's no real reason that difficult terrain would not still be difficult if you had a gust of wind pointed at you, I think the default assumption is that they do stack.

Put another way, speed reduction being capped at half requires that the mechanics of the world be changed in certain instances, and since there's no rule to address it (and it would have been easy to include), I don't think that's how they intended it to work.

Thank you, Scatterbraind. I will take this to my next session and hope the DM will accept the argument.

This blanket statement is definitly incorrect. Movement can cost x2, x3 or even greater and things can also reduce your speed, which is the ammount you can move in a turn, regardless of the movement cost is a given space.

Ask them how they handle the Restrained condition then?

Restrained

• A restrained creature's speed becomes 0, and it can't benefit from any bonus to its speed.
• Attack rolls against the creature have advantage, and the creature's attack rolls have disadvantage.
• The creature has disadvantage on Dexterity saving throws.

Let us know if they have a rebuttal, maybe they don’t understand the rules correctly, which is easy to do (there is a reason there is a Rules and Game Mechanics forums on this site). 

Or even just grappled. Something even players can attempt to do at will, barring size limitations etc.

Yeah, the definition of the grappled condition is literally just that a grappled creature has it's movement speed reduced to zero and how the condition can be ended. Negating movement is all that grappling does.

As others have said, difficult terrain doesn’t stack but things like difficult terrain and Plant Growth or Spirit Guardians that reduces speed, does stack. 

Thank you for your answer. I do appreciate it.