Start a new game with the "Here Be Dragons" origin. Continue through the quest chain, and then ACCEPT the dragon's offer to raise its children.
Accepting this will do two things:
- You will gain access to the Dragon as a playable unit (1 Dragon)
- You unlock the starbase component "Dragon's Hatchery", that gives you the ability to create younger dragon units (10 Dragons)
After this is fully upgraded and all are produced, you will have 11 of 13 dragons.
There are several ways to get 2 more, but each requires some luck:
- Find the Ether Drake. If you play with the "Reanimators" civic, defeating the Drake will allow you to resurrect it, giving you 1 Dragon. If you put a mining station on the "Dragon's Hoard" planet, then you will eventually (wiki says under 50 years, it was a lot longer for me) get the option to raise a young Ether Drake. Accept this for 1 Dragon (1-2 Dragons)
- Unlock the L Cluster, and hope you get the L-Drakes outcome (this is relatively rare, I've gotten 2 times). If you have access to L-Gates, this can reward multiple drakes (up to 4 Dragons)
- Find Shard. Unlocked from the "Kleptomaniac Rats" archaeology site. Reanimate Shard if you have the correct civic (1 Dragon)
- MAYBE: Use the Prethoryn crisis and obtain the Prethoryn Relic from the Queen. This will spawn a single fleet on cooldown that might count for the achievement.
A bigger map will have more options, but you need to get lucky with one or two of the options above. The easiest way (if possible), would be to get the young dragons and beat the Prethoryn Scourge. This also takes a lot longer than either the L Cluster or the Ether Drake.