That was fast.
Gonna go with ones he performed or covered, as well as originals.
Fuck, there's too many. Maybe a Soundgarden list, then a Cornell list?
Top Ten Chris Cornell w/Soundgarden Songs
10) Been Away Too Long (King Animal)
9) Let Me Drown (Superunknown)
8) Black Hole Sun (Superunknown)
7) Burden in My Hand (Down on the Upside)
6) Head Down (Superunknown)
5) Slaves and Bulldozers (Badmotorfinger)
4) Rusty Cage (Badmotorfinger)
3) Fell on Black Days (Superunknown)
2) Outshined (Badmotorfinger)
1) Jesus Christ Pose (Badmotorfinger)
Top Ten Chris Cornell Songs (My Personal Faves/Performances)
10) Cochise ( w/Audioslave)
9) Redemption Song (Bob Marley cover)
8) Like a Stone (w/Audioslave or unplugged solo)
7) One (U2 music w/Metallica lyrics)
6) You Know My Name (Bond theme from Casino Royale)
5) Fell on Black Days (acoustic Soundgarden song)
4) Call Me a Dog (w/Temple of the Dog or unplugged solo)
3) Seasons (Singles Soundtrack)
2) Hunger Strike (w/Temple of the Dog)
1) Nothing Compares 2 U (Prince cover)
Short Answer: Honorable Mentions: Billie Jean (Michael Jackson cover), Mission 2000 (from Mission Impossible:2), Blow Up the Outside World (Down on the Upside), Live to Rise (Avengers Soundtrack), Better Man (Pearl Jam cover), everything else on Superunknown and Badmotorfinger.
Note: I ended up omitting his entire solo catalog. It's too much, and needs its own dedicated list. If you believe, as I do, that there was quality in everything he did, go check those albums out.