Is this the place to talk about rigid analytic geometry?
Beilinson’s article on the crystalline period map reviews some log and Hyodo-Kato theory, and gives nice references. http://front.math.ucdavis.edu/1111.3316
Etale cohomology of rigid analytic spaces
Faltings cohomology, Fontaine-Messing cohomology, Finite polynomial cohomology, Formal cohomology, Monsky-Washnitzer cohomology, Hodge-Witt cohomology??
Log theories:
Hyodo-Kato cohomology, Log Betti cohomology Log convergent cohomology Log crystalline cohomology Log de Rham cohomology Log etale cohomology Log Hodge groups Logarithmic cohomology Logarithmic Hodge-Witt cohomology Relative log convergent cohomology
Niziol has written about K-theory of log schemes.
An interesting question is that of a six functors formalism in p-adic cohomology. There is something by Daniel Caro on arxiv. One could consider Cisinski-Deglise style realization using the rigid cohomology spectrum. Christian Johansson mentioned some potential application (Fujiwara trace formula?????).
Syntomic cohomology, Rigid cohomology, Rigid syntomic cohomology, Modified syntomic cohomology, Relative rigid cohomology
nLab page on E30 p-adic cohomology III