Contra III's Stage 4 consists of a Battleship Raid against Red Falcon's gigantic airship. How is this stage awesome? When you attack the airship's core, your helicopter launches a series of missiles with you climbing on top of each missile as you attack the core.
The final battle with the Relic of Morai in Shattered Soldier and the final battle with Master Contra in Neo Contra.
In Hard Corps, we get GTR (or Super Power Robot Yokozuna), who happens to be a large blue humanoid robot that has some skilled attacks. However, there are two things that just add up the awesome factor: his intro (he runs alongside the train you're walking on and then goes in front of it, "stopping it with his own hands") and his creative death (he falls off of the train and gets smashed to pieces by said train). The Contra creators must've thought that this was an awesome boss, because a MKII version of GTR appears in Shattered Soldier. That one also stops the train with its bare hands after catching up to the train.