Availability: 30 minute standard zone repop.
The laocoon requires a full 24 RL hours of waiting before it may be done. It can be soloed, duoed with 2 max archons for maximum efficiency, but with anything less, a form of 3 or 4 helps a lot. At least 1 fighter must have Mark of Courage and Mark of Wisdom. This boss is a solo kill, so do not think you can power through it quickly.
Preparation
Before the laocoon is unlocked, you must go N, U, N, E from Sigil recall and say something to the white-maned man. The druid will tell you about her treating him, but she ran short on money to complete it. It is up to you to spot her ~1 million gold. Once she gets the remainder, she will complete his therapy in 24 RL hours. Return to her when that time passes to get a password.
Take the usual preparatory measures. Check mems, reagents, calm, fatigue and high explosive mixtures if you have an herbalist are always helpful too.
Head down to Palace of Diocletian and make your way to the catacombs. From the entrance, go 4n w n e 3w nw sw s 2w n until you reach a barrel-chested cultist deep within it. You'll know you're going the right way when the volume of monsters drastically decrease. Sayto blind cultist <password you just received> to gain entry into the beginning of the Laocoon's area.
Scan around this area and look into rooms before moving for a hidden monster named Black Fijuez. Defenses should either be set to dodge or auto/shield block as the spider can both inflict bite paralysis and cast fire/icebolts. After killing it, you'll get a key. Climb down the wall and unlock the door deeper into the catacombs.
You'll encounter a series of rooms with tiles marked with letters. Be careful here as a wrong step leads to instant death. Look into the various exits and check which letter is on which tile. You want to cross this area stepping on tiles that spell out 'O-R-G-A-N-A' in that order. Once you reach the last 'A' tile, exit north and you'll reach the stone tiger.
The stone tiger has a heal script that goes active if the person fighting it lacks either Mark of Courage and Wisdom. Your fighter(s) must have both marks to defeat it. If you only have one combatant with them, have him/her solo the tiger and flee south to the rest of the form for heals. You cannot heal the fighter from out of combat as by just being in the tiger's presence w/o both marks activates his rapid healing.
The Laocoon encounter
Once the tiger dies, you're clear to fight the laocoon. It is located in a private CPK room and must be soloed. The laocoon can rapidly cast a number of highly destructive spells, including web, silence, enervation, energy orb, extinction, firebolt and icebolt. By "rapid casting," it can usually complete these spells in 2 seconds, so take extreme caution and try to limit spellcasting to when it doesn't cast. The laocoon can also occasionally lash with its tail and breathe frost. Afflicting it with fatigue helps limit how often it breathes and lashes.
It is important to note that YOU MUST NOT DIE IN THE LAOCOON'S ROOM. If that happens, the laocoon is instantly restored to full health. If you are near your limits, flee south into the death trap, revive and go back down. Have the next available fighter go in and chip the Laocoon down. Therefore, the biggest threat is getting webbed and silenced. Herbalists can have an easier time surviving in such a situation with calm herbs.
Once the laocoon dies, it respawns in a half hour. A portal opens up leading back to the main catacomb area. Use the portal as if you die, you lose the key from Black Fijuez. You may not camp his room as when the laocoon respawns, it casts an instant kill spell when you engage him in combat so don't get cpk-ed. This is to prevent duoing what a supposed to be a solo boss. Mark of Ministry is awarded to those with Mark of Wisdom and a positive align. Be sure to loot fast and good luck.