What are the adverbs and the adjectives in the following? "much wholesome" and "any other"

(It's not actually homework, I am actually studying for a competitive exam.)

The following sentences have been taken from Wren and Martin and I am providing their solutions for figuring out adjectives:

He gave the boys much wholesome advice

Now Wren and Martin says much is a different adjective and wholesome different, shouldn't it be just wholesome in superlative degree? and can two adjectives come one after another? shouldn't one become an adverb because now it's modifying an adjective?

Lead is heavier than any other metal

Now again Wren and Martin says any and other are two different adjectives, shouldn't other be the adjective and any adverb?

No. Advice that was "wholesome in superlative degree" would be "very wholesome advice"; "much wholesome advice" means "a great quantity of wholesome advice". The word "much" modifies the noun "advice" (or maybe the noun phrase "wholesome advice"). Similarly, the word "any" modifies the noun phrase "other metal".

Both "any" and "much" are adjectives of quantity. They modify nouns; both can modify comparative adjectives (as in "much heavier", "any heavier"), and "much" can modify verb participles when they're acting as adjectives (as in a "much used frying pan") but they do not modify regular adjectives.

Neither any nor much are adjectives. Neither is other, either.

They are Quantifiers, a type of Determiner, the same part of speech as (for instance) the, this, twenty-three, and most of the rest of. Determiners have very complex syntax in English -- an unbelievable number of the questions here are about them -- but they didn't feature much in Latin, so the Roman grammarians didn't notice them at all.

In fact, the Roman grammarians didn't even notice Adjectives, which weren't in the original Eight Parts of Speech; since Latin adjectives inflect and behave just like nouns, they were simply treated as nouns, and the extra POS slot was filled by "Participle" (a category which also accounts for a lot of ELU.SE posts).

Anyway, neither quantifiers or determiners made it into The Big Eight Parts of Speech, as touted by Big-Time Grammar Books, until quite recently. The sentence quoted in the question has no adverbs, but wholesome is certainly an adjective, modifying advice.

For a competitive exam in English grammar, what matters is not English grammar, but what the teacher or official who wrote the exam thinks counts as "English grammar". Since there are some people who will believe anything about English grammar, it follows that what we think or reply here is irrelevant.

For instance, if the question is posed by someone who believes that much "is an adverb, because it modifies an adjective", then the question is closed, for that exam. That doesn't make it a correct answer, but it makes it the right answer on that test.

Sorry.

In both sentences the use of two adjectives together is perfectly grammatical. There is no rule that two adjectives can't be used one following the other. In another example you would have:

I bought a yellow second-hand shirt.

Both yellow and second-hand are two different adjectives and placed one after the other, both qualifying the noun shirt.

EDIT: To refer to your sentences more specifically, you can have much advice and wholesome advice separately, but you can also have much wholesome advice together. In this way you have more information about the word advice in one sentence, since both adjectives qualify the noun.

The case is similar with your second example.