Should one adjust p-values for multiple testing corrections in research? I have seen some medical research papers which do various testings and none of them will adjust p-values accordingly. A lot of p-values are p-values of testing some simple hypothesis.
Why people do not report adjusted p-values for multiple comparisons in those research papers with tables after tables of p-values? I would assume that the any test done in paper is not equivalent to some other test done in the paper and they are not testing nested hypothesis.

The researcher may only be interested in controlling each per-comparison error rate rather than a particular family-wise error rate.  If the researcher makes it clear that many hypotheses were investigated then the reader can interpret each p-value with a grain of salt or perform their own multiplicity adjustment.  Additionally, the p-value may be interpreted as the weight of the evidence for competing hypotheses rather than used in an accept/reject framework.  Even with strict control of a family-wise error rate, no hypothesis is proven false with a single small p-value nor is it proven true with a large one.
The challenge with multiplicity adjustments is defining the appropriate family of hypotheses.  Should the family be restricted to only the "primary" hypotheses in an experiment?  Should it include all of the hypotheses investigated in an experiment?  Why should the family be limited to only those hypotheses in a particular experiment?  Should the family include hypotheses from other experiments to be investigated (or previously investigated) by the experimenter?  Taken to an extreme, multiplicity adjustment will force the experimenter to retain nearly every null hypothesis.
In my opinion the only sensible approach is to interpret the p-value as the weight of the evidence.  Any decision or conclusion drawn from the results is tentative.  This may be the approach taken by some journals.
