Ads pop up and play automatically, daring readers to shut them down with feats of fine motor control. The ads commandeer the screen. They expand and contract. They cover the text and refuse to budge.

And then there is the dreaded X — the one that invites you to close the ad yet seems impervious to repeated clicks of the cursor or the jabs and thrusts of even the most powerful fingers. (Perhaps you have tried a hammer?)

Sometimes the ads dance and move across the screen, forcing the user into a hot pursuit of the X.

“How many times have you hit the X and it doesn’t work?” said Tony Weisman, the chief executive of the digital agency DigitasLBi North America. “Now it’s just a cruel joke.”

Online advertisers and consumers have tried to outmaneuver each other since the early days of the web — with sellers continually finding ways to prolong engagement with ads and users trying equally hard to avoid them. But the cat-and-mouse game has reached a critical point, especially as devices have gotten smaller: Ads have become so annoying, consumers and industry executives say, that they could sink the Internet if they were not also helping support it.