The Cottage in Evony Age 2 can be built from level 1 to 10 and as the name suggests, it increases population limits.
Each upgrade of the cottage will increase your population limit as per the below table.
|3||T. H. 2||400||2000||400||200||00:05:00||600|
|4||T. H. 3||800||4000||800||400||00:10:00||1000|
|5||T. H. 4||1600||8000||1600||800||00:20:00||1500|
|6||T. H. 5||3200||16000||3200||1600||00:40:00||2100|
|7||T. H. 6||6400||32000||6400||3200||01:20:00||2800|
|8||T. H. 7||12800||64000||12800||6400||02:40:00||3600|
|9||T. H. 8||25600||128000||25600||12800||05:20:00||4500|
|10||T. H. 9||51200||256000||51200||25600||10:40:00||5500|
NOTE: You need a Michelangelo Script to grow a level 9 building to level 10
My recommendation is not to have more than 6 to 8 level 9 cottages per city. Cottages take up building slots which are more useful when Barracks are built on them. Besides, the extra population is going to do no good to you unless you are taxing your city heavily thereby reducing loyalty.