Devil May Cry 3- sums up Dante's character and the tone and appeal of the franchise quite well. This party's getting crazy! Let's Rock!
And for an old school one:
Metal Gear 2- the first Metal Gear game that feels like a Metal Gear game with a complex plot and a cinematic presentation which is evident from this opening sequence. The theme song is also really catchy.
Street Fighter 3: 3rd Strike- I like this opening mostly for how extremely catchy that theme song is.
