Is it grammatically correct to say "Many more happy returns of the day "?

Solution 1:

It is not redundant because "many" and "more" do not mean the same thing. It is simply longer. "More" and "longer" or "better" for example is not grammatically correct because they both serve the same meaning, but many and more do not mean the same thing.

"More happy" is not grammatically incorrect, but the "many more" refers to "returns" not "happy"

Solution 2:

There is nothing ungrammatical about the first sentence. Adjectives can, and often do, accumulate in front of nouns. I admit that the construction does sound odd, but you have probably heard people say 'much more', which is equally odd, albeit more common.

It is, however, not redundant. To simply say 'many happy returns of the day' implies that the addressee has not yet received such a happy return, whereas 'many more happy returns of the day' implies that he has received at least one and still has more to come. The two words, though they play similar roles, need not be used exclusively of each other.

As for the second sentence, it is most definitely redundant--but is it ungrammatical? I know of no rule that proscribes repeating oneself. If we compare it with other common phrases that employ double adverbs, such as 'I like him very very much' and 'this soup is really really good', or with those that employ double adjectives, which Shakespeare often used, such as 'most best' and the like, then we see that it really is a common thing to do. The usage is redundant and can be considered lazy, but I see nothing ungrammatical about it.