Low-depth arithmetic complexity of the product of $k$ matrices
I am not sure about specifically depth-three lower bounds, but there has been a lot of depth-4 (and 5) lower bounds, usually assuming other constraints as well. For instance (and without any claim of exhaustivity):


Hervé Fournier, Nutan Limaye, Guillaume Malod, and Srikanth Srinivasan. Lower Bounds for Depth-4 Formulas Computing Iterated Matrix Multiplication. 
Suryajith Chillara, Partha Mukhopadhyay. Depth-4 Lower Bounds, Determinantal Complexity : A Unified Approach.
Suman K. Bera, Amit Chakrabarti. A Depth-Five Lower Bound for Iterated Matrix Multiplication.