How to easily overstuff your barns. 
Note: this does not work for FC animals.
1  First, make sure that your barns are full, (includes a super shed) set the farm you are using as your default.
2 Click on your barn and click on view animals. When this opens, you'll be given an option to "move" your animals. 
3 Click move, and your animals will be put into storage. This is the only time you will have to do this. 
4 Next, enter offline mode, then right click on Play and open the game in a new tab. Once the tab is open, click to go offline again, and right click play and open game in a new tab  again, and again, go offline.  Repeat this process until you have as many windows as you want opened. 
5  Next, go back to you first farm, open up your storage, select "items" and then from the drop down box, select "animals" and then click the option to move them into your barn. Repeat this step in all of your open tabs. 
6 Once you have the last tab finished, close all but the last tab - on the last tab, click your play button to refresh the game and return to online status. 
7 Open your barn, and you should see that is overstuffed.  Happy Farming Everyone!
Below are some images from some of my animal sheds, from different farms, as an example.