Can you elaborate on that? I feel like I've missed something.

There is no "deep logistic regression" because such a construct is just called a "neural network". (It's not the only kind of neural network however.)

Linear regression on top of linear regression with non-linearities in between is a neural network.

Neural networks are just networks of logistic regressors.

or, even worse, useless, because linear transformations are closed under composition.