instance Monad m => MonadPlus (MaybeT m) where
    mzero     = MaybeT $ return Nothing
    mplus x y = MaybeT $ do maybe_value <- runMaybeT x
                            case maybe_value of
                                 Nothing    -> runMaybeT y
                                 Just value -> runMaybeT x

The last line is wrong. It should be, "Just value -> return value". But that doesn't cause the problem.

instance Show (MaybeT m a)

This is never valid. You've defined show, shows, and showsPrec in terms of each other, creating unbounded recursion. Delete it.

*Main> askPassword
Loading package transformers-0.2.2.0 ... linking ... done.
*** Exception: stack overflow

This triggers the unbounded recursion, when it tries to show askPassword. Note that there is no way to show IO values, so there's no way to show MaybeT IO values.

Instead, use runMaybeT askPassword