Question: What type of quality combinations exist? In TF2, there are Strangifiers which can add the Strange Quality to an item. I have seen Strange Genuines, but what other types of combinations exist?

Answer: 
This is a difficult question to answer, because the different qualities that are possible for weapons are [specific to each weapon with almost no rhyme or reason](https://wiki.teamfortress.com/wiki/Item_quality/Possible_weapon_qualities).

You can apply a [Strangifier](https://wiki.teamfortress.com/wiki/Strangifier) to any quality type, so you can have Strange Vintage, Strange Collector's, Strange Unusual, or Strange Genuine weapons/miscs. You can also apply a [Killstreak Kit](https://wiki.teamfortress.com/wiki/Killstreak) to any quality weapon as well, so you can have Strange Killstreak Vintage and Strange Killstreak Collector's weapons. However there are no Unusual quality items that also have a Killstreak kit (since Professional Killstreak Kits are essentially Unusual-ifiers for weapons).

It gets down to what kits you can apply to an item. The qualities that cannot be "applied" are Genuine, Vintage, Unusual, and Collector's (I'm not counting the less-common ones like Haunted, Self-Made, Community, etc). From that base quality, you can apply Strangifiers and Killstreak kits.

And the names just stack up. If you apply any kind of kit, the weapon just gains a new name. This is demonstrated by my Wicked Nasty Professional Killstreak Southern Hospitality. I'd like to go find a Collector's weapon that can be Strangified and Killstreaked.

EDIT: Cosmetics can now have professional killstreaks! Therefore there are hats that can be Unusual Professional Killstreak or even Strange Unusual Professional Killstreak, I believe.
Well, if you're asking about items, which have 2 types of qualities (Like Genuine Strange, which you already mentioned), yes, there're some, and there isn't many of them. Here's a list of such items.:

* **Strange Collector's:** PBPP/Cow Mangler/Air Strike/Righteous Bison/Phlogistonator/Manmelter/Third Degree/Widowmaker/Pomson 6000/Vaccinator/Classic/Cleaner's Carbine/
* **Strange Genuine:** Widowmaker/Righteous Bison/Anger/Camera Beard/Merc's Pride Scarf/Foppish Pschycian/Archimedies
* **Strange Vintage:** Camera Beard/Stockbrocker's Scarf

A-a-and that's all. Most of others Genuine and Vintage items don't have strangifiers for them, and I haven't mentioned **Strange Haunted** and **Strange Unusual**, because there're tons of them (You can check on market).