Not a bad list (though I don't think any quest mods are really "must have")
My list would probably be:
SkyUI -mandatory!
Alternate Start -I like the opening, but after playing it dozens of times, it's nice to be able to skip it
Immersive Armors -probably my favourite mod
Immersive Creatures -I prefer this to Higher Level Enemies since that mod sort of turns the game into Oblivion's leveling system
Apocalypse Spells -adds some much needed spice to the magic system
All of the Unofficial Patches -mandatory!
Quality World Map -being able to see the roads clearly is something I could not do without now
Run For Your Lives/When Dragons Attack -never again will your favourite shop keeper die after trying to fight a dragon
Honestly, I like the base game a lot, but after trying to think of what I consider the mods that I couldn't do without, I realized that there's a lot more than 8. I know a lot of people criticize Bethesda for not doing a lot of these things, but the reality is that if they had spent the extra time developing all of the amazing mods there are, the game probably never have released. I still don't know what they were thinking with the interface though...
Anyway, here's a long list of honourable mentions:
Better Vampires
Moonlight Tales
All kinds of graphics mods, pick the one you like best. Climates of Tamriel is a good starting point IMO.
Interesting NPCs
Cloaks of Skyrim and Winter is Coming Cloaks
Frostfall -one of my favourites, but since it has such a big impact on gameplay, I couldn't call it mandatory (similarly how I wouldn't call realistic needs and diseases mandatory)
Wet and Cold
Either UFO or AFT
Dance of Death
Realistic Needs and Diseases
SkyRe -bears mentioning since it's definitely a good mod, but since it overhauls a large amount of the game, I couldn't call it mandatory. To be honest, I'm not really a huge fan of it, especially since it conflicts with so many other mods. Still, a lot of people swear by it so it's definitely worth a try.
ReProccer if you're going to use SkyRe
Immersive Patrols
Lanterns of Skyrim
Better Dialogue Controls -This one almost made into my top 8 since it's such a fantastic quality of life change.
Unique Uniques
The Paarthunax Dilema -FUCK YOU DELPHINNE!
Inconsequential NPCs
One of the combat overhaul mods -I prefer Duel Combat Realism
Dual Wield Parrying and the animation mod that goes with it
Deadly Dragons
Stranger Danger
TESV Edit -some people swear by it, I haven't really needed it. LOOT has been good enough for me. Speaking of witch...
LOOT -not really a mod, but it optimizes your load order and lets you know if something is potentially going to cause issues. This is the replacement for BOSS.
There you go. There's others that I like a lot, but I wouldn't consider that important (admittedly, some of the mods on my list aren't that important either, I just like them a lot). Keep in mind, there's a lot of texture mods that I like as well but I really didn't want to list them all.