Kukalaka was sort of a soft puppet, stuffed with wadded cotton. When Bashir was a boy, he took him everywhere he went. After a few years, he started to get a little threadbare, until one day his leg tore open and his insides spilled out and made a mess. His mother, Amsha, was all set to throw him out, but Bashir wouldn't have it.
In 2346, at the tender age of five, Bashir performed his first "surgery" on the toy, re-stuffing it, and stitching its leg closed. From then on, Bashir did whatever it took to keep Kukalaka together. Over time, Kukalaka would appear to have nearly every square inch of his body either sewn, stitched or patched, Bashir said jokingly. He hung on to Kukalaka well into his adulthood. In 2372, the bear was kept on a shelf in Bashir's room. (DS9: "The Quickening")
In 2373, while Bashir was dating Leeta, he allowed her to borrow Kukalaka when Leeta mentioned she thought Kukalaka was cute. When Leeta failed to return him after their break-up, Bashir enlisted Nog and Jake Sisko to get him back in exchange for five liters of anaerobic metabolites suspended in a hydrosaline solution, though they would not tell him why they needed it. (DS9: "In the Cards")
Kukalaka remained with Bashir, adorning a desk in his quarters. Before leaving for a medical retreat on Casperia Prime in 2374, he told Kukalaka "See you in a few days, old chum. Keep the home fires burning." (DS9: "Inquisition")