For example, ResNet from 2015 had 152 layers.
A real neuron takes in the order of 10ms to integrate and fire to the next neuron. Many subconscious reactions take less than 1 sec, which leaves time to a chain of length less than 100. Note that those neurons are not strictly arranged in layers.
The human visual cortex has 10^12 synapses [1]. One popular 2015 deep learning net (ResNet 152-layers) used 10^12 FLOPs to classify objects in one image (but less weights.)
In terms of depth, we're there. In terms of breadth, it will take several years. But the brain does things very differently. For example, it has top-down signals during "prediction."