Best DLC To Add

(TL;DR I'd personally recommend Sword of Islam, Republic, Old Gods, Way of Life, Horse Lords, Reaper's Due, Monks and Mystics, and Holy Fury. The rest are mostly flavour and random events, but go for it if that interests you.)

Expansions

The Sword of Islam: Adds only the ability for the player to have multiple wives as a character of the Valyrian religion.

Legacy of Rome: Adds castration and blinding capability as well as a couple of flavour events. It does NOT add retinues.

The Republic: Allows the Merchant Republic realms to be played, namely the Free Cities, Meereen, Yunkai, Astapor, Ibben, Asshai and the Guilds of Qarth. These realms will still be active even without the DLC.

The Old Gods: Enables the concubine system for Drowned God, Dornish, Pirate, Eastern characters and others (Pre-DLC system will be used if not enabled). Also adds much flavour, including beserkers, adventurers, new rebels and runestones. Raiding is available for drowned god, wildlings, pirates and others even without this DLC.

Sons of Abraham: Enables the council of the Most Devout and their election of the High Septon. Also adds many religious flavour events, including various Holy Order events/decisions.

Rajas of India: Adds flavour to various far eastern cultures, including tiger hunts, gurus, reincarnation events and other flavour events.

Charlemagne: Adds the chronicle, custom kingdoms (existing titular kingdoms are still available without DLC) and a 'hero knight' flavour event

Way of life: Enables the focus system, allowing a ruler to pick any of the foci available with all associated events decisions. Seduction focus replaces 'visit chambers' interaction when DLC is enabled

Horse Lords: Adds the nomad mechanics for Dothraki/Jogos Nhai (DLC now required to play these), landless adventurer mechanics for Ironborn, the wolf flavour events and land trade routes.

Conclave: Adds the enhanced council/favour mechanics, new education focus system, dynamic mercenaries plus other misc flavour

The Reaper's Due: Adds improved epimedic and illness mechanics, including hospitals and court physician treatment events. Adds prosperity and depopulation, as well as other flavour events.

Monks and Mystics: Adds the devil worshiper societies (needs to be activated via game rules), crown jewell/weapon artefacts and new councillor job actions. Also add the paid quality of life features- allied AI orders and the improved child naming/prisoner interfaces

Jade Dragon: Adds unjust conquest casus belli, chinese portraits/buildings which are used in Yi Ti, and some artifact/chinese flavour adapted for the mod.

Holy Fury: Adds Shattered/Random World system, Warrior Lodge societies, Tiered-scar system, Kill List, Forge Bloodline ambition, Coronation decision, Sway and Antagonize decisions and other misc flavour. Roman portraits used in Asshai.

Other DLC

Ruler Designer: Fully compatible with the mod

Customiser: Fully compatible with the mod

EU4 Converter: Not compatible

Music Packs: The mod's soundtrack currently overrides them, however deleting "A Game of Thrones/music/songs.txt" will re-enable vanilla music

African Unit Pack/Portraits: Used for the Summer Islanders, Basilisk Islanders and Sothoryos

Mongol Portraits: Used for the Nefer and Yi Ti

Turkish Portraits/units: Used for Lengi

Spanish Portraits/units: Used for Moraqi

Celtic Portraits/units: Used for Ibbenese

All other Face/Unit Packs: Not used

Dynasty Coat of Arms Packs: Not used

Sunset Invasion: Some things may happen if enabled...

Also, the newer sub-mods will be more likely to be compatible than the older ones, but I wouldn't use any that aren't listed as compatible with 1.8 or 1.9, due to how major the 1.8 update was.