I feel like the best two typings that have been mentioned here today are Bug/Psychic and Electric/Ice. Both fill the credentials set out by bmb at the start, I'll refer to jas and srk for more info on those because of how well they have explained it, a refresher isn't really in order. One type that has yet to be mentioned so far that I've found I feel fits our credentials is Ice/Ghost. Also kudos to fryfrey, I used your formatting cause it's pretty.
Passive Damage Weakness: Easily Applicable. Ice/Ghost is weak to Pursuit and Stealth Rock, with Spikes and Toxic Spikes as a threat as well. It is immune to Hail, but weak to Sandstorm. In terms of passive damage it really covers a lot of what we consider. It does however resist U-Turn so there is an exception in that sense.
Good Offensive STAB Combo: Easily Applicable. The Ice/Ghost combo hits 6 types super-effectively, Grass, Ground, Flying, Dragon, Psychic, and Ghost. However there is a little issue, it is shut down by Steel types. Although many common steel types, Metagross, Jirachi, and Skarmory come to mind, are neutral against the other offensive type so it balances out fairly well.
STAB that cannot realistically be spammed endlessly for maximum impact: Easily Applicable. Ice is resisted by four types, Steel and Water being the primary two that will quickly stop spamming. Ghost has the Normal immunity giving it trouble, even if Normal isn't the most common typing. As well Dark and Steel shut it down pretty quickly as well. I'm sure it's safe to say that both can't be used endlessly.
Lots of weaknesses and resistances: Applicable. This is rated Applicable because of the typing being weak to 5 types, and resisting 3 and being immune to 2. This seems like a perfect balance but the types being resisted aren't as common as the attacking types. Ice/Ghost is weak to Fire, Steel, Dark, Ghost, and Rock. It's safe to say your opponent will have those types on his team somewhere. The weaknesses here really do amplify the risk on this Pokemon. The resistances are Ice, Poison, and Bug, with a Fighting and Normal immunity. The Ice resist and the Fighting immunity are the best in terms of pros with this typing, although I wouldn't turn down the Bug resist any day.
STAB on notably risky moves: Fair. This is pretty uneven really, it depends on how we're playing the Pokemon. Ice STAB is almost always Ice Beam and Ghost is almost always Shadow Ball. Neither is a risky move, both have excellent accuracy and extra effects. Ice has more options for a risky attacker in Avalanche, Icicle Spear,Blizzard, and possibly Icicle Crash and Frost Breath. Ghost has interesting options in Destiny Bond, Ghost Curse, Hex, and Shadow Force. Overall I would put it as a neutral stat.
So to sum up, Ice/Ghost has a lot of good qualities that we should consider for this CaP, it definitely carries lots of risk but could also have some amazing rewards attached. Bug/Psychic and Electric/Ice would be equally as valid.
Passive Damage Weakness: Easily Applicable. Ice/Ghost is weak to Pursuit and Stealth Rock, with Spikes and Toxic Spikes as a threat as well. It is immune to Hail, but weak to Sandstorm. In terms of passive damage it really covers a lot of what we consider. It does however resist U-Turn so there is an exception in that sense.
Good Offensive STAB Combo: Easily Applicable. The Ice/Ghost combo hits 6 types super-effectively, Grass, Ground, Flying, Dragon, Psychic, and Ghost. However there is a little issue, it is shut down by Steel types. Although many common steel types, Metagross, Jirachi, and Skarmory come to mind, are neutral against the other offensive type so it balances out fairly well.
STAB that cannot realistically be spammed endlessly for maximum impact: Easily Applicable. Ice is resisted by four types, Steel and Water being the primary two that will quickly stop spamming. Ghost has the Normal immunity giving it trouble, even if Normal isn't the most common typing. As well Dark and Steel shut it down pretty quickly as well. I'm sure it's safe to say that both can't be used endlessly.
Lots of weaknesses and resistances: Applicable. This is rated Applicable because of the typing being weak to 5 types, and resisting 3 and being immune to 2. This seems like a perfect balance but the types being resisted aren't as common as the attacking types. Ice/Ghost is weak to Fire, Steel, Dark, Ghost, and Rock. It's safe to say your opponent will have those types on his team somewhere. The weaknesses here really do amplify the risk on this Pokemon. The resistances are Ice, Poison, and Bug, with a Fighting and Normal immunity. The Ice resist and the Fighting immunity are the best in terms of pros with this typing, although I wouldn't turn down the Bug resist any day.
STAB on notably risky moves: Fair. This is pretty uneven really, it depends on how we're playing the Pokemon. Ice STAB is almost always Ice Beam and Ghost is almost always Shadow Ball. Neither is a risky move, both have excellent accuracy and extra effects. Ice has more options for a risky attacker in Avalanche, Icicle Spear,Blizzard, and possibly Icicle Crash and Frost Breath. Ghost has interesting options in Destiny Bond, Ghost Curse, Hex, and Shadow Force. Overall I would put it as a neutral stat.
So to sum up, Ice/Ghost has a lot of good qualities that we should consider for this CaP, it definitely carries lots of risk but could also have some amazing rewards attached. Bug/Psychic and Electric/Ice would be equally as valid.