I've had my players gauge each others equivalent ability score in the real world, and then take an everyday scenario like "you all are hanging out in front of Starbucks..." Usually something catastrophic will happen, or a bit of subtle foreshadowing that leads them into flipping their worlds upside down. With these game I've had them only start with what would normally be on their person at the time.
The longest iteration of one of these games, happened while the group was out on a backpacking trip in Northern California, when minor tremors turned into a massive earthquake that opened a crevice. The players climbed down into the crevice to investigate and find a globe sized crystal, still buried to its top. After they search the immediate area and attempt to dig it out, the crystal begins to emit a white and purple aura, another tremor happens, and they fall onto the globe. They were instantly teleported across our plane and into the Forgotten Realms smack dab in the middle of The Moonwood forest. This campaign lasted about a year.
Most of the time, they would try and find a way back to our universe, but you'd have one or two the players that would be a bit over-zealous and try and strike a claim in the fantasy world.
I've done an alien invasion where they'd have intelligent mounts (a black panther, velociraptor, white lion, and a "chocobo") locate (and scare them at first), then guide them to the safety of the 'good' alien race to better inform them.
Another one of my favorites was that a friend of ours was a super-secret agent who was part of a para-military agency that had its location under his house. This 'friend' had been training the players in real life via Halo where each of them would take control of their own Spartan to do various missions, where in each mission they would go up in military ranks, command other squadrons and platoons.
In regards to the fantasy setting, one campaign, the PC's had dreamed about an event, and would feel an incessant pull to its location until the party would be gathered to stop or witness the event.
An early game of mine, I made the PC's start off as classless, child versions of the race they wanted to play. They were all already friends, and would meet up in the town center to rough-house and play 'stick wars'. Just a simple game where they would either throw or hit each other with sticks, but one of the NPC kids that ran with them accidentally murders another kid with his innate and at the time unknown magical powers. Depending on how they played during the encounter would determine their class.
Those are some of my unique ways of starting campaigns...but I'll usually default and have everyone meet up in a bar and get into a bar fight xD
