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

Shady Trough

Capacity:

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:

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.