My answer to this question is in media res. I vastly prefer to drop the players into the middle of something that's going on rather than drop them into a low-stakes situation and just have them decide where to start, so to speak. What that looks like specifically depends on the campaign, it could be a fight, or it could be something more like the chained in a ship example above (which I loved btw, whoever posted it). I'll usually pick something that reflects the themes of the campaign, which is something I always make very clear in session zero. I like the game to start with some momentum built-in I suppose. Not always, but mostly.
I have an idea to start a short campaign in the midst of 3 different scenes for different characters, each of which is going badly for the characters involved. One pair of PCs and their companion are running from extra-planar modron cyborg cops through the streets of a city on one of Eberron's moons, another PC starts off falling down the side of a tower due to a cut rope as she was betrayed, and the other 2 PCs stand back to back in a the center of a circle of blades pointed at them.
The scenes each lead the PCs to a tavern in Sharn, where they will meet a prospective patron who had been observing their exploits (and had some tricks up his sleeve if anyone had been killed of captured, which would have led to those PCs owing him). The NPC patron is an NPC from the current Eberron Campaign, and the stories of the PCs will tie into the fallout of what they are doing in the current campaign. The scenes are based on the character concepts that I've seen from players, and would change if they go different ways with their characters.
The current Eberron campaign, I used the starter adventure premise from the 4e Eberron campaign guide, with some modifications. The PCs have changed a bit over time, but that game started with 3 PCs and a DMPC, being a sort of reboot of a stalled out Eberron campaign that began in 4e. The 4th PC would be joining later, due to scheduling stuff and the fact that his background involved being elsewhere on this particular day.
The PCs are scouting out a tower on the edge of Cyre, near the border with Breland. 2 PCs are mercenaries working for Breland, the other PC is a "squire" (she was supposed to attain full knighthood but her mentor sent her to the farthest front he could and refused to knight her because she countermanded his orders, so she's a bit disgraced at this point) of the Onyx Skull from Karrnath. The DMPC is a mercenary working for Cyre, though only he knows that ever once they've all met.
The knight is investigating the tower because she is following the trail of her investigation into the Emerald Claw, who had been abducting people for experiments, using magic intended for undead on living subjects, back in Karrnath. The Shadar-kai monk DMPC feels terrible magic building, and volunteered to look into it, thinking some sort of real bad war magic was being used. The other two are investigating sightings of undead being raised, which would indicate some sort of Karrnathi offensive.
As they get to the tower, a storm rolls in and covers their approach, and they catch glimpses of eachother, and before they can confront eachother, they come upon a necromantic ritual being down with a tied up human man in the center, and a prophecy mark on the ground. There is a cool moment where the Brelish mercs see an obsidian sword blade, and black and red plate boot, and a crimson cloak and fiery long hair, and nothing else, in a flash of lightning, because the knight is basically a metal album cover come to life.
They get a flash of prophesy referencing 5 who will stand against the storm, and see themselves standing together against some great darkness, and then the necromancers attack them. Once that's dealt with they free the human and investigate the rest of the tower, find more emerald claw above about to sacrifice another guy, save him, he thanks them and names himself as Brenn ir'Gadden, of a powerful Brelish noble house. The other guy is his long time friend and assistant. Introductions done, the storm worsens, and more enemies arrive.
They hold out against a tide of undead and 3 necromancers with the help of Brenn (appropriately to his characterization, the assistant rolled garbage and I think hit with one bow attack the whole fight), and the storm grew to the point where lighting was constantly hitting around them, and the sky was fully black in what should have been the first light of dawn, and they all instinctively started to run as a hundred foot tall wall of blue-green mist rushed toward them and they heard the sounds of battle in the valley below turn to screams and then go silent. A short scene of them all standing at the edge of the border, staring into blue-green mist unbelieving as they try to grapple with what they'd just witnessed, and then we end the session.
Next session cuts directly to 4 years later, in Sharn, and they've all been invited by Brenn to a remembrance day event on the anniversary of the Day of Mourning. Other PCs that I'd introduced in another session are also there, in the common balconies, taking note of how each of them knows either Brenn or one of the PCs, as well as some memorable NPCs that were either PCs or important NPCs in the old Eberron campaign. The scene is interrupted by an attack by a Mournland abomination, and some terrorists, one of which seems to have an aberrant dragonmark. A chase battle through the towers and skybridges and rooftops of sharn ends in a townhouse with a complex arcane lock, then into a basement where an enormous eldritch machine is gathering power and needs to be disabled, in a complex skill challenge that took 4 characters IIRC 4 rounds to complete, while the rest of the characters chased and caught the aberrant marked terrorist, and discovered that he was Brenn's old friend and assistant.
The fallout from that adventure is still defining the campaign 7 levels later, as they have fully investigated and are now figuring out what to do about a conspiracy between the Emerald Claw, delkyr cultists, and a cult known as The Order of The Black Flame, which is in service to an ancient red dragon, who is also the dragon that murdered the Kobold wizard's gold dragon mentor. Basically all the seeds of the whole campaign were sown in that first two part adventure.
