For years, Ben Black’s phone annoyed his family. It was the only Android device in a family message group with eight iPhones. Because of him, videos and photos would arrive in low resolution and there would be green bubbles of text amid bubbles of blue.

But a new app called Beeper Mini gave him the ability to change that.

Mr. Black, 25, used the app to create an account for Apple’s messaging service, iMessage, with his Google Pixel phone number. For the first time, every message the family exchanged had a blue bubble and members were able to use perks like emojis and animations.

Since it was introduced on Dec. 5, Beeper Mini has quickly become a headache and potential antitrust problem for Apple. It has poked a hole in Apple’s messaging system, while critics say it has demonstrated how Apple bullies potential competitors.

Apple was caught by surprise when Beeper Mini gave Android devices access to its modern, iPhone-only service. Less than a week after Beeper Mini’s launch, Apple blocked the app by changing its iMessage system. It said the app created a security and privacy risk.

Apple’s reaction set off a game of Whac-a-Mole, with Beeper Mini finding alternative ways to operate and Apple finding new ways to block the app in response.

The duel has raised questions in Washington about whether Apple has used its market dominance over iMessage to block competition and force consumers to spend more on iPhones than lower-priced alternatives.