How big is Daenerys Targaryen's army by Season 4? (Game Of Throne)?

Well, she bought all the Unsullied in Astapor (8,000). While sacking Yunkai she got the Second Sons, which I think they mention having around 2,000 soldiers. She did liberate loads of slaves from Yunkai and is now doing the same with Meereen, but those are just women and men/children, so I don't count them as part of the fighting army. She probably has about 10,000 as of now. Her belief is that when she comes to Westeros with that 10,000 (plus 3 dragons, which makes a HUGE difference), a lot of the people will immediately surrender/join her/rally to her cause.

The core of her forces are about 8,000 Unsullied warriors, less those who have died in fighting already. She also has a couple of mercenary companies in her employ, the Second Sons and the Stormcrows. They give her about a thousand more troops. In addition, some of the slaves that are following her have formed sort of ad-hoc military units. But they're untrained and poorly armed and not of much use. The Dothraki who follow her are mostly women, children, and old people. She really only had her Bloodriders, although they're starting to train up some of the older boys among her people