(covers information from several alternate timelines)
Malcolm Reed's birthday cake
Kes' Jiballian fudge birthday cake
A Prime holding a birthday cake
A birthday cake was a cake served to a person on his or her birthday. A birthday cake was often decorated with small candles. Human tradition held that the person with the birthday might make a wish, which would come true if all the candles could be blown out in one breath.
On September 1, 2151, the crew of the Enterprise threw Malcolm Reed a birthday party complete with a pineapple birthday cake. (ENT: "Silent Enemy")
In 2350, Annika Hansen's sixth birthday cake on the USS Raven had six candles on it, and one more to grow on. Ironically, she was assimilated not long after. (VOY: "The Raven")
In 2367, there were six candles on Commander William T. Riker's blue frosted birthday cake at his birthday party. (TNG: "Future Imperfect")
In 2370, in an alternate quantum reality, the staff of the USS Enterprise-D held a surprise party for Commander Worf, with a chocolate birthday cake. During the party, Worf shifted into yet another alternate quantum reality in which the inside of the cake was now yellow. He commented that he thought the cake was chocolate, to which Counselor Deanna Troi, notoriously fond of chocolate, commented "Don't I wish." (TNG: "Parallels")
For Kes' second birthday in 2371, the USS Voyager crew threw a surprise party for her. Neelix made a seven layer Jiballian fudge birthday cake with L'maki nut icing and punch. (VOY: "Twisted")
In an alternate reality, Kes also got a birthday cake for her ninth birthday. (VOY: "Before and After")
In 2373, Miles O'Brien's birthday celebration included a birthday cake, which was actually made by a Pah-wraith that, at the time, was possessing his wife, Keiko. (DS9: "The Assignment")
In 2376, Janeway replicated Tuvok a birthday cake with a candle on it. He claimed that this was a Human tradition, and Vulcans did not "blow out candles." In the end, he complied, claiming that "it was a fire hazard." (VOY: "Fury")
In 2380, a Prime renting an event silo after Clar had a birthday cake for his daughter's 100th birthday. (LD: "Veritas")