One of the NPCs in my campaign has been captured and the next session is primarily focused on breaking them out. He's been captured for roughly 8 or 9 days and in that time has not had a long rest (he has several levels of exhaustion on him). He is a spellcaster, though, and I'm not sure how to address how many spell slots he should still have. He probably used a good chunk of them trying to escape capture, but I'm not sure if I should leave him a couple or just remove all his spell slots completely. Thoughts on how I should calculate this? 

It depends on how many slots he used before being captured and during his captivity. Another factor is the spells at his disposal. If he doesn't have spells to help him escape, there's no reason in wasting slots, but he could preserve it. NPCs don't have to be super strong and know everything, but that doesn't mean they're stupid.

Only you will know how many spell slots he has. The exact number isn't really important because you can add or subtract whatever you need to fit the situation as it unfolds. A better question is, how do you plan to use those spell slots? 

If it's to help the party because they're under-leveled and could use an NPC's firepower to survive, then maybe keep a couple spells in reserve. If it's for verisimilitude, it's perfectly reasonable that a guy who has been held captive and treated poorly to the point that he's on his way to death by exhaustion would have zero magic left. Maybe his captors forced him to cast spells, or maybe the extreme lack of sleep made him crazy, casting stuff at hallucinations. Maybe you give him all his spells precisely so he can attack the party because he thinks they're going to hurt him, not save him. Whatever fits the story and makes it fun for your players is the right answer. 

I'd say he has none. The prison should be aware of the existance of magic in the world, and either have a Magical Lightning Rod (touch it and it drains your charges) or a large orc with a cudgel to convince new captives to cast any remaining slots into the open. And a way of checking whether they're out of slots - our table always assumed Detect Magic would allow this, but it may be a hold-over from 3.5.

Quote from Acromos >>

I'd say he has none. The prison should be aware of the existance of magic in the world, and either have a Magical Lightning Rod (touch it and it drains your charges) or a large orc with a cudgel to convince new captives to cast any remaining slots into the open. And a way of checking whether they're out of slots - our table always assumed Detect Magic would allow this, but it may be a hold-over from 3.5.

 If you have such things, expect your players to get and use them.

Also, cantrips make it impossible to run a caster out of spells.

Holding mid-level casters prisoner is hard in D&D. You can stop a lot of spells by removing material components, but you're likely going to have to keep their hands bound and possibly also gag them. (This would explain the lack of sleep.)

As for the original scenario, it depends a lot on the circumstances of their capture. If they were grabbed in their sleep, they'll have most of their slots. If it was at the end of a long fight, their tank will likely be close to empty. There's also the question of whether they have any useful spells prepared. (Not every caster lives like an adventurer, and wizard in particular are likely to choose their spells for what they expected to be doing that day, which could leave them poorly-prepared when they're suddenly taken captive.)

I think the best answer from a narrative sense is that they'll have a few slots left with which they can help out, but it's going to be the players' show.

I think I'd go with "How much do you want this caster to be helping in the breakout?"

If you design the encounters to be something the chardacters can do on their own you can remove nearly all the spell slots and it's fine. If you want the NPC to participate some, you can put in a few spell slots. Or, and this is what I'd do, wing it. Don't plan a set number of slots. Instead use them, if needed. That way you can flex to what the player characters are doing.

Quote from jl8e >>

 If you have such things, expect your players to get and use them.

 

 What - an orc with a stick? They're welcome to him.

The lightning rod setup requires (obviously) a substantial anti-magical infrastructure - what with the special materials, the deep earth grounding, the runes and wards and so on. It also takes time, thaumic discharge happening at a rate of conductivity multiplied by mass but divided by thaums available (or in laymans terms, 1 spell level pr. 10 minutes, at best). And even then, there are diminishing returns. So it's not something an adventurer could pack in their backpack. But should they desire to build a large prison compound, then yes, certainly, that's a thing they can then add to it. 

It's no more weird than taking away the barbarians axe. Someone, somewhere along the line, figured it would be unwise to put fully armed enemies in prison. This applies equally to casters and martials.

Quote from Agile_DM >>

I think I'd go with "How much do you want this caster to be helping in the breakout?"

If you design the encounters to be something the chardacters can do on their own you can remove nearly all the spell slots and it's fine. If you want the NPC to participate some, you can put in a few spell slots. Or, and this is what I'd do, wing it. Don't plan a set number of slots. Instead use them, if needed. That way you can flex to what the player characters are doing.

 You can indeed make him weaker for the time being of being held captive, give him less HP aswell for being tortured or starved. After he has been freed and gets decent rest to recuperate from his prison time he could shine by doing something back to the group when in dire need. Then the NPC has his spell slots and HP fully recovered.

The lack of abilities that drain spell slots in D&D is an interesting absence. Absent an ability like that, he probably has most of his spell slots, because usually the best way to deal with a caster is to make them unable to cast spells.

Quote from Pantagruel666 >>

The lack of abilities that drain spell slots in D&D is an interesting absence. Absent an ability like that, he probably has most of his spell slots, because usually the best way to deal with a caster is to make them unable to cast spells.

 There's still the 'orc with a stick' option for draining spell slots. Or dropping casters in a cell with Darkness and Silence. 

Or, as I assume would be the case in more barbarous times ... straight up mutilation. We tender, civilised souls may balk at such things, but I'd say it an almost certain guarantee that you're not going into prison with your magical abilities at your disposal. Not unless they're totally unaware you have them.

Quote from Acromos >>

Quote from Pantagruel666 >>

The lack of abilities that drain spell slots in D&D is an interesting absence. Absent an ability like that, he probably has most of his spell slots, because usually the best way to deal with a caster is to make them unable to cast spells.

 There's still the 'orc with a stick' option for draining spell slots. Or dropping casters in a cell with Darkness and Silence.

 That doesn't drain spell slots, it makes them impossible to use. Which means they're still available if at some point you regain the ability to cast spells.

Quote from talisettitulis >>

One of the NPCs in my campaign has been captured and the next session is primarily focused on breaking them out. He's been captured for roughly 8 or 9 days and in that time has not had a long rest (he has several levels of exhaustion on him). He is a spellcaster, though, and I'm not sure how to address how many spell slots he should still have. He probably used a good chunk of them trying to escape capture, but I'm not sure if I should leave him a couple or just remove all his spell slots completely. Thoughts on how I should calculate this? 

 If the caster is captured for several days he had plenty of time to take a long rest and get all its spell slots back. If you ruled he couldn't rest, then he will have whatever spell slot left remaining he didn't use. In the uncertainty, you can always decides a number or determine it randomly ex. 2d4

Quote from Pantagruel666 >>

 That doesn't drain spell slots, it makes them impossible to use. Which means they're still available if at some point you regain the ability to cast spells.

 It does, however, achieve the same goal: Making sure they don't use their magic to either escape or assault the prison guards.

This is one reason NPCs are built differently than PCs. Any given NPC has exactly the number of spell slots you need them to have for a given situation. How much help do you want him to be able to offer? That's how many spells he has.

Don't overthink it. He's not a primary character, he's a plot device.