How can I get an animation to loop a number of times before transitioning to another animation?

Trying to loop a looping animation 5 times before it transitions to the next animation.

Can I do this from the animator? Or does it have to be via scripting?

In the Animator view, select the exiting Transition for the Animation Clip you want to loop, and expand Settings section in Inspector. Change the Exit Time to be greater than 1.0 -
For example; If Exit Time is set to 2.5, it would loop your animation 2 1/2 times.

You could make a for loop and put the trigger of the animation in it, and once it has looped a certain amount of time, you could then trigger the next animation.