A good reason to make something right-associative is if it takes a Foo and a Bar and gives you a Bar in return. In this case, left-associative grouping would only lead to type errors.
Taking a look at Haskell's right-associative operators, the cons operator (:) seems to follow that rule, and others seem to take into account other idiosyncrasies. Here are my guesses for each operator:
Exponentiation to the power of a natural number (a ^ b ^ c), an integer (a ^^ b ^^ c), or a float (a b c) is right-associative, probably to match the parentheses-less math notation: (a (b c)).
The function type constructor (a -> b -> result) is right-associative (a -> (b -> result)) so that it's easy to use multi-parameter curried functions. For the same reason, function currying/application (func a b) is left-associative ((func a) b).
Low-precedence function application (a $ b $ c) is right-associative. This might be because (a b c) tends to be sufficient for passing the usual curried parameters, so (a $ b $ c) is reserved for cases where the output of each ($) isn't another function to apply. If this is true, ($) usually takes a function and a non-function and returns a non-function, so it's naturally right-associative.
Haskell has a few other right-associative operators out of the box, but I'm not sure why. In some cases, maybe it helps with perfomance.