How many languages did Tolkien invent?
This is not as straight-forward a question as it first appears. What do you mean by language? If you mean something usable for normal conversation, then possibly none of Tolkien’s inventions contain sufficient attested vocabulary to qualify as a language (though a case might be made for Quenya and/or Sindarin). If you mean some set of words that Tolkien named (e.g., even the Black Speech, with only a couple dozen attested words), then there are many, many languages. And if you allow revisions of a language to count as a language in its own right (Tolkien revised his work many times over many decades), there are many, many more. In other words, this question may be frequently asked, but it is rarely answered, simply because it isn’t clear what the answer to such a vague question ought to be. See Helge Fauskanger’s essay on this topic for a more detailed answer.