As of the moment, our game is only limited to 55 Animals that we can place on the Farm. Limits are being controlled by the Troughs that we place on our Farm. In addition, the number of Troughs that we can have are also limited for now. We can only have up to 6 Water troughs and 1 Shady trough that can be purchased from the General Store. Each Water Trough will give us additional 5 capacity of Animals from the Farm while the Shady Trough will give us additional 10 capacity of Animals from the Farm. Here's the information that we can use in maximizing our Animals using our Troughs and our Animal Buildings:
Water Trough




1stFree Purchase Milk Meadow Expansion+5 Animals
2nd 5,000 Coins Level 3+5 Animals
3rd20,000 Coins Level 12+5 Animals
4th28,000 Coins Level 21+5 Animals
5th45 Farm Bucks  - NA -+5 Animals
6thFree  Old Watering Hole Expansion (Level 68)  +5 Animals


Shady Trough
1st  10,000 Coins     Level 5                                                     +10 Animals


Initial Limit (Start of the game): 10 Animals
Water Trough: +5 Animals
Shady Trough: +5 Animals
Upgrade to Deluxe Shady Trough: +5 Animals
To avoid confusion, 10 Animal capacity is given to us at the start of the game. Adding Water Troughs will give additional 5 (6 max Water Troughs = 30 capacity) and adding a Shady Trough will give us additional 10 capacity. If you upgrade your Shady Trough to Deluxe Shady Trough, additional 5 capacity will be added. All in all, we can only have 55 maximum animals roaming on our Farm. 
Selling Animals
You can also sell your Animals to make more space for others using the Sell tool in the Tool Menu:

User-added image

Note: The price you get for selling your Animals is only a fraction of what you paid to get them. The profit for your Animals lies in keeping them around to produce Resources that you can sell, rather than selling the Animals themselves.