Does a metric exist that quantifies morphological richness in languages? Either a numerical score, or at least a ranking of languages would suffice.
No, but as contrasted with the related notion of morphological complexity, it seems at least a bit more possible to define and count it. When people say a language has a "rich" morphological system, that usually means that there are very many non-lexical morphemes. It might also mean that there is the possibility of combining affixes, but I don't think that is inherent in the notion of richness. If a language has 200 affixes (a reasonably rich morphology), it would be unusual if e.g. there were 200 cases or mood suffixes, and no other morphemes. Simple counting would be a good measure of "richness".
Complexity is much harder to define, since it necessarily involves rules of combination. For example, instead of just saying "the past tense marker is -ile", you might have to say "the past tense marker is -ita after the causative -ʃ and -ile otherwise", which makes the system more complex. The problem is that there is no scientific utility to having a metric of "richness", that is, unlike water which boils when it reaches a specific temperature, languages don't have properties that depend on having a particular quantity of richness.