They definitely can go deeper - with skip connections and normalization you can propagate gradients through any depth of architecture.
Adding more layers isn't free though, it requires more parameters and thus more compute. There's an optimal depth-to-width ratio for a given parameter count.
Exactly the answer I was looking for, thank you!
It becomes very expensive compute-wise, but where we are actually running up to the edge is the scale of the data. They've discovered "scaling laws" (see chinchilla paper) that determines how big your model should be given the amount of data you have. We could go bigger but there's no reason to use a multi-trillion parameter model for example because it's just wasted capacity.
The bottleneck is the total compute budget devoted to training, so while I'm quite certain that stacking a few more layers can be done and would have some benefit, it might well be that spending the same extra compute on a larger context window or 'wider' layers or simply doing more iterations on the same data would have a larger benefit than more layers, and if the people training the very large models think so, they would do these other things instead of stacking more layers.