Law of excluded middle in MLTT Is it possible to add law of excluded middle to Martin-Löf type theory as an axiom? It seems to me, that it's possible to add it to Coq since Coq has a module for non-constructive reasoning. Also, it seems to be impossible to Agda since it leads to inconsistency. But can we add it to plain MLTT? And why it will/won't work?
Law of excluded middle does not make intuitionistic logic inconsistent; intuitionistic logic is in many regards a subset of classical logic (of course classical logic can't have proof objects and there are other diffrences). In Agda you can use the postulate keyword for such things. The following could be added to your postulates.

$LEM : \{A : Set\} \to A \lor \neg A$

I was doing some proofs out of a book on first order logic where I found it more useful to have double negation elimination than LEM.

$\neg\neg E : \{A : Set\} \to \neg\neg A \to A$

One of course implies the other so take your pick.

edit:
One of your previous questions gives some insight here as well. cody made mention of Agda being predicative. Martin Lof type theory is predicative (as well as Agda which is basically MLTT). Agda still has large elimination and adding LEM does not make the logic inconsistent. Martin Lof originally tried to base MLTT off of System-F but its impredicative nature kept getting in the way of such things. This is part of the reason why MLTT is predictive.