I'm going to introduce Ice/Fire, myself.
Passive Damage Weakness: Easily Applicable. Sandstorm-weak, Spikes-weak, Toxic Spikes-weak, and quad-Stealth Rock-weak. Also severely weak to some priority moves - but, notably, not to Bullet Punch, and with a double resistance to Ice Shard.
Good Overall STAB Combo: Easily Applicable. This really is an absolutely insane STAB combo. Ice is resisted by Fire, Steel, Water, and Ice; Fire is resisted by Water, Rock, Dragon and Fire. There are only two types that really can be said to get a safe switch-in on these STABs; Fire and Water. Beyond that- Dragons can switch in on a Fire move, but will die if they eat an Ice one; the same applies to Steels, but in reverse. For a risky Pokémon, this is fantastic; these are two of the most common types in the metagame, and whether they switch in for free or eat huge amounts of damage depends on your prediction abilities. If you know the opponent's party is running Genesect and Hydreigon - do you use Fire Blast or Ice Beam? Offensively dangerous while creating a dimension of risk from the get-go.
STAB that cannot realistically be spammed endlessly for maximum impact: Easily Applicable As mentioned, these STABs can't break through Water or Fire types; this means that a good bulky Water can check the Pokémon nicely. You get almost, but not quite, perfect neutral coverage in trade for the weaknesses this typing gives you.
Lots of weaknesses and resistances: Somewhat applicable Ice/Fire gives you resistances to Bug, Grass, and Ice; Grass isn't a big deal, but Bug lets you come in on U-turn and Ice on one of the most common types of coverage moves in the metagame. These are usable openings. On the other hand, you have a laundry list of weaknesses - Ground, Rock, Water, and Fighting. These are also common offensive types; you need to find just the right moment to switch in, but that moment WILL come.
STAB on notable risky moves: Easily Applicable. Blizzard, Avalanche, Icicle Spear, Overheat, Eruption, V-Create, Flare Blitz, Heat Crash - Ice/Fire provides an excellent set of powerful offensive options that can come back to bite you if you take a bad bet.
Basically, the thought here is that it's a phenomenal offensive typing with a few important resistances supplementing an otherwise bleak defensive picture.