Swallowing a Spider

"I know an old lady Who swallowed a bird ... She swallowed the bird To catch the spider ... She swallowed the spider To catch the fly But I don't know why She swallowed a fly ''I guess she'll die"

- A Nursery Rhyme

Lensman Arms Race is an escalation of countermeasures or "bigger sticks" between two or more conflicting parties, which can be frustrating enough. But some people manage to inflict this on themselves by "solving" problems in a way that creates some greater problem instead... and if they still can, do it repeatedly.

Summon Bigger Fish is particularly prone to this.

Digging Yourself Deeper and Revealing Coverup are subtropes of this.

Film
"James Rolfe: The good news is, Godzilla's back; and the bad news is, Godzilla's back."
 * Kiryu from the films Godzilla Against Mechagodzilla and Tokyo SOS is an attempt by the JSDF to create a weapon out of the skeletal remains of the 1954 Godzilla. They succeed, but, unfortunately, Kiryu tends to act like his flesh-and-blood counterpart.
 * Likewise, in the 1991 film Godzilla VS King Ghidorah, people in Japan attempt to "re-create" Godzilla so that they can stop King Ghidorah (Long story short, Time Travel has prevented a Godzillasaurus from becoming...well, Godzilla.. and...let's just say it gets confusing....). Not only do they succeed in mutating a Godzillasaurus into Godzilla (Don't ask how they even know what Godzilla is considering he was "erased from history" in the film), but they also end up making him bigger AND more powerful than before. And....then Godzilla proceeds to rampage across Japan yet again...


 * Syndrome's Omnidroid in The Incredibles was the most recent iteration of his "super-killing" robot. It was intelligent and competent enough to realize that it didn't want to be commanded or shut off -- and powerful enough to be able to insist on it.

Literature

 * Guard-bird by Robert Sheckley. So we got learning robots to help law enforcement. Their learning evolved out of control and now they rampage through the countryside? Let's - this can only end well!

Western Animation

 * The Simpsons: when lizards ate the pigeons Skinner began to plan the next step (Chinese needle snakes to eat the lizards). And the next...

Real Life

 * Somebody imported and released European rabbits in Australia. They reproduced, well, like rabbits - and without adequate competitors and predators they became a problem. Then someone stumbled upon a solution: let's fix this by releasing some European foxes. Three guesses as to what happened next.