WhatsApp, Apple iMessage, Facebook Messenger, Google Hangouts: There’s a good chance you’re already using two or three different messaging apps just to keep up with friends and family. Alphabet Inc.’s Google wants you to try one more, Allo. And it is worth a look, even if you never text with another human: Do it to strike up a conversation with the Google Assistant, the search giant’s artificial-intelligence helper.

Like Apple Inc.’s iMessage and Facebook Inc.’s Messenger—and nearly every other messaging app out there—Allo...