In Portal, Hoopy the Hoop was a hoop fallen from the sky among the wreckage of GLaDOS at the end of the game. Hoopy landed on the Aperture Science parking lot in front of Chell just before being dragged off by the Party Escort Bot.
As stated in an interview with Valve on "Opening The Portal: Exploring The Game's Development", they never expected "The Cake is a Lie" to become such a huge hit. The developers expected, if anything, that Hoopy the Hoop would become the meme derived from Portal.
Erik Wolpaw explains, "We thought we should have a warehouse full of Hoopy t-shirts and mugs and posters…we would watch that hoop roll by over and over again. That was the part of the game we were most proud of, and nobody cared."
Hoopy the Hoop rolls across the screen at the end of the game as Chell lays outside the Aperture Science facility.
Hoopy is known to appear in a few places within Portal 2. Its earliest known appearance is in GLaDOS' ruined chamber, where it lays slightly to the left of what could be described as GLaDOS' head; its second known appearance is underneath a floor panel in Test Chamber 02. It then makes it's third known appearance behind a desk in Rattmann's den in Test Chamber 06; this is the same den as referred to by the Final Transmission achievement.
Hoopy's fourth appearance is in multiple custom test chambers (Created and published with the Perpetual Testing Initiative.) It is seen by noclipping into the ceilings of the exit rooms, although it doesn't appear in each one.
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Hoopy the Hoop at the end of Portal
Hoopy the Hoop in Portal 2: Lab Rat.