AT&T Inc. played it safe with its new streaming service HBO Max by selling it for $14.99 a month, the same price it currently offers for HBO, and making it free for current HBO customers.

The online service, which will launch next May, is essentially a supersize version of HBO. Besides HBO programs, it will have hit shows and movies from the Warner Bros. library as well as new content including shows from prolific TV producer Greg Berlanti and actress-producer Mindy Kaling.

It...