Credit goes to the dudes who came up with these combos.
-Dwarf Barbarian: Fatigue regeneration means you can berserk quite often, and crafting kinda makes up for not starting with oils of transformation.
-Human Druid: A jack of all trades with some of the most overpowered abilities. At level two you can see through fog, so you can cast Obscurement and spam summons from relative safety. There are some decent damage spells and a lot of utility ones. You can use animal bond to summon a mount that can help you fly, see in the dark, and move really fast. Wild Shaping unlocks a lot of the class' potential, including a late-game form that makes you practically invincible. You also start with decent melee and ranged weapons.
-Human/Drow/Elf/Gnome/Dwarf Mage: Similar to the Druid in some ways, this class is generally even more specialized in casting spells. You can also use Obscurement and summon spam, along with better damage spells, and starting out as a Weavecrafter lets you control teleport effects, which is extremely useful.
(Gnome Mages make good Illusionists. Dwarves can use Metamagic abilities quite often.)
-Lizardfolk Monk: Not the most powerful class in the game, but they can hold their own in combat and learn a lot of skills. (Turn Natural Attacks on in the options until you get some Monk-centric feats to make Martial Arts more effective.)
-Human/Orc Priest of Asherath (Domains: Strength, War, and either Knowledge or Time): Self-buffing priests are pretty dang powerful. Orcs are a good fit for this class because of inherent Toughness giving them a lot more HP. Asherath is chosen for his domains and because he is not easily angered, unlike most other gods.
-Human/Elf Ranger (archery): One of the best combinations, you can shoot endless arrows from atop a fast, flying, and infravision-granting mount. I favor the bow and arrow style because arrows are way easier to find than other types of ammo. Oftentimes you can shoot arrows from such a range that the enemy doesn't know where you are.
-Drow Rogue: A combo that fits together easily. The Drow can cast Globe of Shadows at clevel 2, which vastly boosts the stealthy playstyle. Levitation is quite handy at times. Infravision, good stats, and ambidexterity are icing on the cake. Not to mention the fact that Rogues get a ton of skill points to throw around.
-Human/Orc Warrior: There are a lot of options available to the player based on perk and weapon choice. Firstly, there is the super-good spiked chain build; play as a Human and take the Exotic Weapon, Monkey Grip, and Two-Weapon Style feats. Eventually you will be able to dual-wield spiked chains, thanks to the oil of transformation you start with.
Another good combo is playing as an Orc with a shield and a good one-handed weapon. You can go with the Tough Fighter feat tree to become a rather durable tank. You could go with dual-wielded scimitars (for an excellent parry score) or two-handed weapons as well. All of the warrior feat trees are pretty nice, too.
You can also play as a charisma-centric character. Inspire fear in your enemies and gain allies as you descend.
Some mighty multiclass characters:
-1 Warrior/10 Priest (of Asherath): Start out as a Warrior for those precious oils of transformation, then proceed as a Priest as normal.
(Note: Starting as a Warrior for your first level then switching to another class is good in general.)
-3 Priest (of Asherath; domains: Strength, War, and Knowledge)/Another class: This is a generalist-type of multiclass; you gain just enough levels to cast Identify, then switch to something else, such as a Barbarian or Warrior.
--7 Ranger/4 Druid (or 6 Ranger/5 Druid): Get to 7 as a Ranger for a nifty feat, then switch over to be a Druid. Why? Because additional Ranger levels past 7 don't give you very much benefit, while Druids gain extremely useful abilities. You continue to get better Animal Companions, too.
-3 Mage (Illusionist)/8 Rogue: There's some synergy in the two classes, allowing you to get your sneak attack bonus quite often.
-3 Priest (of Zurvash; domains: passion, pain, strength)/8 Barbarian: Fatigue regeneration without having to play as a Dwarf, plus the pain domain lets you recover health when you dish out damage. Zurvash is relatively easy to follow.
-10 Rogue/1 Mage: Go for a high luck score and cast Mage Armor.
-10 Druid/1 Mage (Weavecaster): You get most of the benefits of being a Druid while also being able to control where you teleport to.
And finally, Prestige Classes. Most of them don't seem to be worth their requirements, but here are a few exceptions:
-3 Mage (Illusionist)/8 Assassin: Quite similar to the Illusionist Rogue.
-1 Warrior/5 Diviner/5 Loremaster of Asherath (daggerspell, dual daggers): An interesting combination. I haven't tried it yet myself, but it seems good in theory.
-3 Mage (Weavecrafter) or 3 Druid/8 Alienist: For those who wish to really push the summoning playstyle.
-3 Mage (Divination)/8 Loremaster: Focus on debuffs with a high intelligence score so they rarely get resisted. Polymorph dragons into rats, and such!
-3 Monk/8 Tattoo Mystic: An obvious progression. Try to become a Tattoo Mystic at level 3 because their benefits scale greatly with your character level. Unfortunately, I haven't played very many of these so I can't comment on recommended tattoos.
Random notes:
-Racial bias: Yes, I really love humans and mostly dislike small-sized races. Small races have a lot of disadvantages I usually don't wanna deal with. Regarding humans...it's funny because I dislike playing as humans in most games with other races because I find humans boring and plain-vanilla, but they are extremely good in Incursion, in my opinion.
Extra skills and feats can make certain combos much more survivable. For example, some builds don't have the Healing skill, which often means getting diseased/poisoned is a death sentence. Searching is a useful skill that can help you avoid traps, and thus further deter status effects. Decipher Script and Knowledge (Magic) are extremely good for helping you identify all of your items (I usually play with a decent luck score.)
Lockpicking is just plain nice. Especially at level 1 the extra experience is quite good. Then there's general great skills like Jump, Swimming, etc.
Regarding stats: I almost always use the point-buy option and Charisma is usually the stat I gut all the way down to 3. I also bring luck down to 3 when playing as a Druid; their starting kit is good enough to win the game on its own. I generally play with decent stats around, otherwise, unless I'm playing as a class that demands higher stats in some respects (strength for a Warrior, etc.)
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So yeah, I figured this mini-guide might be helpful for people coming back to Incursion. Feel free to post some of your own powerful combos!