Four layer PCB VIA drill pairs
Ask your assembly house what is possible.  

However, even if they can do it, it will be at a massive premium to the point that I would bet that HDI stacked microvia with a load more layers would be cheaper.

The issue is that you would need each layer to be drilled independently, and then lined up, they will get lots of duds and it will be a hugely more expensive and time consuming process then say a 8 layer simple or even an 8 layer with one or two layers of stacked microvia (Which is a standard process).

The mix of blind and buried is what makes it a real pain, but blind L1-L2 AND L1-L3 is a pain in the toot, could you not simply use L1-L3 and deal with the resulting stub?

This is however a question for your board house, capabilities vary widely.