Looking at historical missions with the constraints you gave:
- Docking in low earth orbit is a precursor to leaving low earth orbit.
- This rules out space station mission, Apollo-Soyuz, shuttle missions that captured a satellite (like the Hubble repair flights) and the like because they didn't leave LEO.
- Craft from multiple launches are docked.
- This eliminates the docking of the Apollo command and lunar modules after the trans-lunar injection burn (which admittedly means it was already leaving LEO).
- Leaving LEO is to go do something useful.
- Your comments imply you mean to exclude Gemini 11 because it didn't go very far (only 1368.9 km at apogee). Wikipedia claims this is still the record for apogee (because Apollo wasn't in the sphere of influence of Earth at its farthest point) and in terms of absolute manned distance it's only beaten by the Apollo moon missions (which already don't count for the purposes of the question).
Considering Gemini 11 is still the highest Earth orbit and Apollo is the only manned program to leave Earth orbit your constraints have eliminated all historical manned missions.
To take the question a step farther I'll relax the manned requirement: Are there any missions launched in two+ launches that docked and went to do something useful?
The answer appears to still be a solid "no":
Wikipedia lists the very few docking mission types that weren't between a crew vehicle and a habitat (or other crew vehicle) but suggests that two commercial companies have proposed satellite servicing vehicles to refuel satellites in the geosynchronous belt or offer high-delta V orbital maneuvers. Unfortunately, the two proposed vehicles were Mission Extension Vehicle (last news post from August 2014 suggesting they have a government contract but no mention of concrete plans for development or launch) and Space Infrastructure Servicing (last news I could find was from September 2012 with news of a contract but no indication they've flown anything yet).
There was actually a test flight for autonomous non-cooperative docking in 2007, but that doesn't lend itself to missions where docking is planned and doesn't seem to have progressed past the single flight.
Overall it appears that even with probe missions there have been no attempts to launch parts separately and dock them in orbit for the purpose of going to do something useful.
Back to the question in the title. To state the obvious: they'll only be first if they actually fly the mission including an EOR as planned and nobody else beats them to it.
That said, considering this list (taking moon exploration to be the soonest thing that might include EOR for manned flight) they appear to have a very good shot at being first if they can even come within a few years of meeting their schedule of flying by 2018.
There are proposed Orion missions that I believe count on EOR, but considering Orion isn't even scheduled for manned flight until EM-2 in 2021 (which is a single launch) and anything more bold than that is only proposed (and subject to politics) I wouldn't be concerned with Orion actually managing EOR and going somewhere any time soon.
Considering that the ESA Aurora Program seems to have no concrete plans that leaves China with their possible manned moon landing by the mid 2020s. No mission details to suggest if it would be using EOR, but it's also distant enough that Space Adventures could suffer years of schedule slip and still be first.