You want 3M double-sided film, er, # 7935 laminating adhesive. However,
this results in boards that are ∗more∗ flexible, not less. Epoxy is
better, but you need a way to hold the boards aligned ∗and∗ compress
them together while the epoxy cures.
To stick the boards together, I drill out two matching vias on each
board, in opposite corners, and thread brass wire through them. That
way they "just line up" when you press them together. er, you have to
stick the tape to one of the boards first, so you drill out the tape
too. Then peel the other protective film off and stick them together.
For 4-layer boards, you want really thin outer layers (I use 5 mil) and
16-32 mil or so inner (two 16SS or one 32DS). To solder to the inner
layers, you need to pre-drill the outer layers with even larger holes,
so the inner layers are exposed enough to solder to them. You still
need via wires, though, even when connecting to the plane on the same
side, as solder has too high a surface tension to just blob it.
I usually squish a little solder paste into the oversize holes, so the
inner layer has solder and flux before I try to connect to it.
example 4-layer taped-up board:
http://www.delorie.com/electronics/usb-gpio/