What is a "15-hit kiss"? In Assassination Classroom, season 2 episode 15, we see Nagisa Shiota perform a "15-hit kiss" on Kaede Kayano (aka Akari Yukimura) to help save her life. Afterward, everyone is amazed that Nagisa performed a 15-hit kiss, but what is a "#-hit kiss" and what does the number symbolize?
The #-hit kiss is a technique first seen in action by Irina Jelavić in chapter 8 ("Breast Time") in the manga (page 17), or episode 4 ("Adult Time") in the anime (6.5 minutes in), when she kisses Nagisa for a 30-hit kiss.



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It is described by Nagisa in the manga: 


  I truly felt the weight of her "killing" words and that she was the real deal, a professional hitman. A kiss so incredible that it overpowers the senses. It could bring even the strongest men to their knees.


So this kiss is a weapon for seduction and assassination, and Nagisa recognizes its power. This is the reason he knows how to use it on Kaede Kayano later. It's because he experienced it first-hand by Irina.

The hit count is, as a comment mentioned, probably a reference to combo hits in videogames. From Wikipedia:


  The first game to count the hits of each combo, and reward the player for performing them, was Super Street Fighter II.