With current (ME3) technology FTL travel is very demanding. Element zero FTL drives accumulate a static electrical charge when a vessel has been in FTL flight for some time. This charge steadily increases with the amount of time a vessel spends in FTL. Eventually, it must be discharged. There are no places to discharge this energy in the dark space between galaxies. The Reapers knew this of course, that intergalactic travel was not possible for organic civilizations.
Yes, we've been discussing this issue for more than a year now 
You're wrong to state that intergalactic travel is not possible for organic civilisations, though. Reaper technology does not suffer the same technological limitations, which means the current civilisations could achieve "unlimited" FTL travel through a number of means:
1. Independently inventing the same technology, either truly coincidentally (which could looked contrived) or because the Reapers are proof that it is possible. There are many examples throughout human history of what's called technological diffusion, whereby the concept of something - but not its specifics - spreads to new areas and provides inspiration for an independently new invention. Even if the civilisations of this cycle haven't solved the FTL issue, there's zero evidence that this is true for earlier cycles. Technology isn't magic, and if the Reapers can do something it must be possible for others as well.
2. Reverse-engineering actual Reaper engines, either through research on the remains of Sovereign or other Reaper wrecks discovered at some point in galactic history. If we wanted to push the development of unlimited FTL travel back a few years, it could be achieved by either the batarians experimenting on the Leviathan of Dis, or Cerberus' research on the 'dead' Reaper in ME2.
3. Jerry-rigging a discharge system that would survive a centuries-long journey. The game's lore, interestingly enough, never actually says how often drives need to be discharged, nor how much energy is created. The codex say that space stations have facilities for ships to discharge into, which implies that a planet isn't necessary. I've suggested before that all a ship would need to do is discharge the electricity into specialised areas or materials, which are then dumped into space. Build enough of them around a ship to last for the journey (and remember, the time between discharges hasn't been established yet) and you've got a workable if inelegant way for the ship to survive an FTL trip to Andromeda.