I use Polish notation here, where "C" indicates a conditional which is an operator of two arguments.  The formation rules go:

1) all lower case letters with or without numerical subscripts are formulas.  

2) If "x" and "y" are formulas, then Cxy is a formula.

3) For the present purpose, only strings which are formulas according to 1) and 2) are formulas.

I'll assume that if the deduction theorem holds, then the system has CpCqp (Simp) and CCpCqrCCpqCpr (Frege) as theses ("theorems" in the object logic).  If that assumption holds, you only need to find logical calculi where either Frege or Simp do not hold, and the deduction theorem will fail.

Now let's concentrate our attention on axiomatic systems **A** where the axiom(s) are tautologies in classical propositional logic, and the only rule of inference of any system belonging to **A** is condensed detachment "D" (perhaps we could allow ordinary substitution of variables and ordinary modus ponens here and things will still work as follows).  Consequently, we can generate as many (countable) systems where the deduction theorem fails as we want *from a single thesis of classical propositional logic* (though not necessarily any thesis of classical logic, since, for example, (CCNppp, D) has only one thesis).

The axiom I choose here is CCpqCCqrCpr (Syll) (plenty of others will do also!).  Syll holds for Lukasiewicz's 3-valued logic, but Frege does not hold for such a system.  Consequently, Frege fails for the entire system (Syll, D).  But, since Frege fails for (Syll, D), Frege will also fail for (Syll', D) where Syll' is a thesis obtainable in (Syll, D).  Thus, any system (Syll*, D) will not have the deduction theorem.  How many systems (Syll', D) exist?  Well, the variable "r" in Syll does not appear anywhere in Syll's antecedent Cpq (and every thesis of syll is of this type).  Thus, given countably infinite variables, we can observe the sequence (Syll, CCCCqrCprsCCpqs, ...) where any thesis x after Syll is obtained from D(Syll).(x-1) (if x=1, then we have D(Syll).Syll, if x=2, then we have D(Syll).(CCCCqrCprsCCpqs), and so on).  Thus, (Syll, D) has countably infinite theses, which, with the above implies at least countably infinite systems where the deduction theorem fails.

Lukasiewicz-Wajsberg 3-valued logic, BCI, BCK, relevant calculi, Lukasiewicz infinite-valued logic, and equivalential calculi constitute some of the systems studied where the deduction theorem fails (or has a modified form).