The only Pokemon without an elemental weakness are Sableye and Spiritomb, both Ghost/Dark. However, there is a price for this. A lack of any elemental weakness was the entire gimmick behind Sableye, and to balance this its stats are lacking. Spiritomb does the same thing, only with better stats, though its speed is still quite low. Call Spiritomb an effective wall. Good for spending turns trying to Hypnotize enemies, then recover whatever HP lost with Dream Eater. Its low HP is a caution too, keep in mind.
I lol heartily at the notion that Spiritomb could recover any significant amount of HP with Dream Eater. What is your obsession with that sucky move and why would you want to use that (which also requires you to put the opponent to sleep) instead of Pain Split or Rest, both of which will help Spiritomb recover HP more efficiently and without the need for any support moves. Oh, and did I mention they work with Will-O-Wisp too? But yeah, let's not halve the opponent's attack, use a combo of two moves which both whittle the opponent's HP away in a very frustrating manner (Will-O-Wisp + Pain Split) or have something that lets you get back all the HP your opponent has laboured to dent and ditch all status conditions to boot (Rest). Yeah, let's just all go with the wimpy psychic move that barely gets any super effective hits, drains a dismally small amount of HP the few times it
does hit something, and requires you to rely on Hypnosis to even get it in. All together now, yay Dream Eater![/Sarcasm]
Seriously, I'm not going to let you get away with a single moment of giving that move the praise it doesn't deserve. xP
As for the only one or no weaks (yes, I know most of these have probably been mentioned, so sue me):
Ghost/Dark (no weaks)
Poison/Dark (Only Ground, no weaks with Levitate)
Electric (only Ground, again no weaks with Levitate)
Bug/Steel (Fire weak only, no weaks with Flash Fire)
Water/Ground (Grass weak only)
Steel/Flying + Flash Fire/Heatproof/Volt Absorb (Only electric or fire weak)
Normal (only fighting weak)
Dark/Psychic (Only bug weak)
Psychic/Steel + Flash Fire/Heatproof/Levitate (Only fire or ground weak)
Don't think there are any others. If you want an immortal pokémon then it's Ghost/Dark with Wonder Guard and status-ditching moves. x3