Fights where he underdog almost cracked an upset but fell short?

Basilio against Gavilan
Tiberi against Toney
Bill Brennan & Firpo against Dempsey
Floyd Robertson against Sugar Ramos
Keith Mullings against Raul Marquez
Inoue's last opponent

 

Chavez Jr came very close to stopping Sergio Martinez in the final round.

As I look into it though, it looks like Chavez Jr. wasn't as big of an underdog as I remembered.

 

Whitaker vs Hurtado

 

A Norfolk fighter named Pernell Sweet Pea Whitaker, in his prime and at his prime weight, attempted to take the P4P throne from a grizzled, over-the-hill and overweight Julio Cesar Chavez. Sweet Pea fought the fight of his life, a gallant effort in which he pulled out every gimmick and trick of the non-combat realm and even showed flashes of actual combative wizardry. But alas, he could only hold on to force a draw. Still, a great accomplishment, though.

 

Yori Boy vs. Trinidad.

 

No, in fact because he was undefeated in 41 or so bouts and had a world title, this was a super-fight back then.

 

Meldrick Taylor vs Chavez Sr I
Carl Williams vs Larry Holmes