Is “today” a compound word?

Yes, today is a compound word, but a very old one. It wasn’t originally formed from the preposition “to” and the noun “day”; rather, it originates from their Old English equivalents, “tō” and “dæġe.”

In the past, it was sometimes written as a hyphenated compound: “to-day.” But the hyphen is no longer included; it’s always “today” now (“to day” is also wrong).