Is "embiggen" considered a formal or slang word?

Solution 1:

This now appears in the following dictionaries.

https://www.merriam-webster.com/dictionary/embiggen

https://www.collinsdictionary.com/submission/535/embiggen

Here is an article explaining why.

http://www.theweek.co.uk/92117/the-simpsons-embiggens-english-with-official-new-word

Solution 2:

"Embiggen" is not a word I would use in formal communication. It was introduced as a joke on an episode of the television series "The Simpsons," and even in that fictional universe the authenticity of the word is questioned:

"'Embiggens'? I never heard that word before I moved to Springfield."

"I don't know why. It's a perfectly cromulent word."

Solution 3:

C.A. Ward, "New Verbs", in Notes and Queries: A Medium of Intercommunication for Literary Men, General Readers, Etc, volume 10, page 135, pub. 1884:

Are there not, however, barbarous verbs in all languages? ἀλλ' ἐμεγάλυνεν αυτοὺς ὁ λαός, but the people magnified them, to make great or embiggen, if we may invent an English parallel as ugly. After all, use is nearly everything.

Given that it is cited as a word as far back as 1884 (pre-dating what we would consider modern slang), whether it became commonly used enough to appear in a dictionary seems irrelevant to the question. While its main use today is modern slang, it is, or at least has been, a word used in at least some parts of the country in normal conversation.