I'll start with a few
Tyrion- dwarfism, missing part of nose from fight in Battle of the Blackwater, Manson Moore cut him
Jaime- hand cut off by Vargo Hoat
Theon- various body parts cut off by Ramsay Bolton
Lollys Stokeworth- born a dimwit
Ilyn Payne- tongue removed on orders of Mad King Aerys
Donal Noye- lost his arm in a war, maybe Robert's rebellion?
Bran- paralyzed from being thrown out a tower window by Jaime
Strong Belwas- stomach scarred from self inflicted "first wounds"
Drogo- became a "zombie" after being resurrected by Mirri Mazda Duur
Maester Aemon- age related blindness
Hodor- born a ddimwit(as far as we know in the books)
Timmet son of TTimmet- lost an eye by his own hand
Sandor- face burned by his brother Gregor
Ghost of High Heart- born a little person
Ser Robert Strong- lost his head as proof of Gregor's death
Varys- castrated as a boy by a dark magician
Qhoron Half Hand- lost 3 fingers fighting a Wildling
Arya- blinded by the kindly man
The Waif- looks like a child as a result of being poisoned
Lady Hornwood- chewed off her own fingers because Ramsay starved her
Myrcella- lost an ear, caused by Darkstar
Catelyn- became undead, face half rotted, throat cut by Raymond Frey
Penny- born a little person
Groat- born a little person
Davos- missing first joint of fingers on left hand courtesy of Stannis
Jon Connington- greys calendar
Shireen- face half hidden by greys calendar, got it from a toy?
Patchface- became a dimwit after almost drowning
Jon Snow- burned hand fighting a wight
Euron Greyjoy- might be missing his left eye
Wyman Manderly- ate too much and became "too fat to ride a horse"
Doran Martell- gout, can't walk because of obesity
Last couple are jokes :
I guess I got more than 20