Rachel … I apologize if what I say is stuff you already know, but I think steve.carney might be assuming more background knowledge than I have, and some readers will have even less than I do. Hopefully, he’ll correct anything I get wrong, including on pricing. [Yes I’m being “wrong on the internet” in hopes of drawing out a better answer.]
If cost-effectiveness is important, I think the first choice is just a plated through-hole, with antipads (etched-away copper at least as large as an annular ring would be) on any layers that should not connect.
It does traverse (and use space) on all 6 layers, and there will be some stubs where signals intended for layer 3 can go on to layer 6 and bounce back. But this is the most basic drill hole you can get, and anything else goes up to at least the “let a salesman contact you” level.
If the stubs are a problem, you can backdrill, for extra cost. I think this is reasonably standard, but it does seem to move into “let a salesman call you back” territory, so I’m guessing the job becomes about 20%-30% more expensive. (Anyone willing to correct me, please do.)
It sounds like you might also be able to use controlled depth drilling. This sounds a lot like backdrilling. But for reasons I don’t understand, this is the first I’ve heard of being able to use controlled depth drilling to create a blind via, so there might be additional complications that make it more expensive in practice.
Also note that since you would only be able to plate the hole from one side, you might also need more space on the top level for larger holes to get the right aspect ratio. Freeing up space on layers 4-6 at the cost of using more space on layers 1-3 is likely a bad trade.
If you do need to route over that space on the 6th layer, or even the 4th for a 1-3 drill, then I guess you do need blind vias. Hopefully, you could do that with controlled depth drilling rather than micro-vias or sub-assembly/sequential lamination. And if you need microvias anyhow for other reasons, it could make sense.
But if leaving an intact power plane is the reason you’re going even to laser drilling, let alone to multiple lamination, then I think you’ve discarded the cost factor. This is well into “contact us for a quote” level, but I got the impression it added a digit to the price. It isn’t clear how much of that is for the drilling itself, vs the tighter tolerances and registration that usually come with it.