AquaKairi first meets Aqua as a child. Though they only have the one meeting, it has a strong impact on the both of them. One, Aqua gains a new keychain due to the encounter, Destiny's Embrace (which is later Kairi's default Keyblade); secondly, Kairi inherits Aqua's Keyblade, and thirdly, Aqua casts her infamous protective charm to keep Kairi safe. It's unlikely Kairi remembers much of her encounter with Aqua, presently - if she were to see her, her memory might try to fill in the blanks, but given she was so young and Kairi had amnesia for such a long time it would have little to fill in those blanks with. However, Kairi is Aqua's successor and that's quite a title to have, given Aqua was a Keyblade Master, and the only one who became a master of her trio. I would like to see the two of them interact again once Aqua is saved from the world of Darkness; or even for Kairi to be the one to save her. There might not be strict interaction between them, but Aqua has been of great importance to much of Kairi's life, even if Kairi is unaware of it.
Axel/LeaIt's unclear if Kairi knew Lea whilst she lived in Radiant Garden, but I would imagine so. Known or not, Kairi officially meets Axel during the course of Kingdom Hearts II. Wanting to find his best friend, Axel decides to use Kairi to force Sora to co-operate with him, but this isn't entirely successful; Kairi flees from him the first time she gets the offer, and when he forcibly takes her from Twilight Town, Saix comes along to kidnap her instead for more malicious intentions. However, Kairi's relationship with Axel is not just being kidnapped this one time. There was clearly a duration spent in Betwixt and Between, and, if we look solely at the manga, whilst Kairi stands up for herself (and bites his arm in order to escape, have I mentioned the manga does so many things right by Kairi?), she does listen to what Axel wants and realises he is not a bad person; he just wants to find someone he cares about. As Kairi puts it, their goals are the same, and she agrees to go with him in order to help him, showing just how kindhearted she is, even if she only later finds out he was going to hurt Sora. This side of their relationship is, unfortunately, unique to the manga. We never see Kairi forgive him in game and they have no further interaction, save Axel telling Sora to apologise for him... though once more, there is a promise of it. As Kairi is now training in the Mysterious Tower, it is likely Lea will be put forward as her sparring partner. If a friendship or a team is to be developed, I would most like to see it between these two.
SelphieSelphie is a year younger than Kairi, and often hangs out with her respective trio: herself, Tidus, and Wakka. However, in the year whilst Sora and Riku were away and lost, Selphie spent a lot of time with Kairi and they developed a close bond. Whilst not strictly the best of best friends, they are definitely good friends and clearly enjoy each other's company. I doubt there is much on the horizon for the two of them, but Selphie is a dear friend to Kairi and someone she trusts and looks out for, and vice versa.