Why can't there be more hydrogen ion concentration in acids or lesser hydrogen ion concentration in bases? Why should there even be any hydrogen ion concentration in alkalis? (I know pH is calculated by the negative log of H+ ion concentration.)
It isn't. We're just used to working in the lab where solutions are generally 1 M or less in concentration. So we get no more than 1M (solvated) hydrogen ions where pH is 0, or at the other end no more than 1 M hydroxide ions corresponding to pH 14.
These 1 M limits are, however, just for our convenience and not any law of nature. Right in our own labs we likely have concentrated acid stock solutions, well above 1M, where if we attempted to measure pH we would get less than zero. Acid pickling solutions in the steel industry are concentrated enough to do the same thing, as manufacturers are more interested in getting hot mill scale cleaned off than in a convenient pH scale. Bases like sodium hydroxide also have concentrated stock solutions with pH above 14.