For reasons of dramatic tension, I want to have an enemy spellcaster (or monster) that is going to cast a really big, really bad (for the players), spell that the players need to disrupt in some way before it goes off in like 3 rounds or so.
Are there any examples of this kind of long casting on any spell list, monster stat block etc.?
Or is this something that you just need to homebrew for your dramatic storytelling elements?
I can't think of any single spell or ability that takes 3 rounds to complete, but what you're really looking for is a storytelling that device that helps moves the action forward.
That device is called a clock, and you simply just need to create the scenario and encounter you want, but keep track of the 'casting', which really can be any narrative tool you want (spellcast, ritual, trap triggering, etc.) where if the clock fills up, then X happens.
Each round, you just fill a clock space, or not, depending on what the players do. Maybe they slow it down, maybe they stop it, maybe they let it go because they're focused on something else.
But with the clock, you're able to create the scenario that works for your story.
When filtering spells on D&D Beyond, the filter jumps from 1 action (6 seconds) to 1 minute. There is nothing in between. If we assume that the filters on D&D Beyond are working correctly, that means that there is no spell with a cast time between 1 action and 10 actions.
There are some alternatives however.
While Delayed Blast Fireball takes 1 action to cast, its damage increases each round.
So, the sooner the party stops the enemy, the better.
Once again, the spell has been cast. But the sooner the party breaks concentration, the less they will suffer. Call Lighting is a great example as it has a clear indication that the spell is still ongoing.
This is a story telling workaround, but the party could stumble upon the enemy when it is in round 7 of casting a 1 minute spell.
The Dungeon Master's Guide lists Siege Equipment on pages 255 and 256. These weapons typically take multiple actions to fire. For example, the Ballista and the Cannon both take three actions to use.
BALLISTA
Large object
Armor Class: 15
Hit Points: 50
Damage Immunities: poison, psychic
A ballista is a massive crossbow that fires heavy bolts. Before it can be fired, it must be loaded and aimed. It takes one action to load the weapon, one action to aim it, and one action to fire it.
Bolt. Ranged Weapon Attack: +6 to hit, range 120/480 ft., one target. Hit: 16 (3dl0) piercing damage.
Of course, you won't find a Cannon in the lair of a monster. It implies intellectual enemies. If you prefer, you can flavor a siege weapon to be more fantasy-like. The DMG even suggests as much in the stat block for the Cannon (emphasis mine):
CANNON
Large object
Armor Class: 19
Hit Points: 75
Damage Immunities: poison, psychic
A cannon uses gunpowder to propel heavy balls of cast iron through the air at destructive speeds. In a campaign without gunpowder, a cannon might be an arcane device built by clever gnomes and wizardly engineers.
A cannon is usually supported in a wooden frame. Before it can be fired, the cannon must be loaded and aimed. It takes one action to load the weapon, one action to aim it, and one action to fire it.
Bolt. Ranged Weapon Attack: +6 to hit, range 600/2400 ft., one target. Hit: 44 (8dl0) piercing damage.
While others have answered the question you've asked, there seems to be another solution to the problem you're trying to solve. The spell could require material components which are located around the play space, and the components would need to be brought to a brazier or circle or cauldron to cast.
Alternatively, if you don't want your players to be able to interfere with the delivery of the items, they could be being delivered by a lowering platform, which moves slowly because it is driven by a descending weight and a geared mechanism in the cieling.
