I was thinking one day, and was trying to describe some of the latest events in my life. To my dismay, I could not think of any word to describe becoming better friends with someone (in a platonic way, not a romantic one). Is there such a word?
Relationships that become stronger are said to deepen.
Escalate is often used to characterize the act of increasing the intensity of something.
Bonding comes to mind, usually used in a familial context, but sometimes for friendships as well. It would seem odd to apply it to a romantic relationship, unless that relationship is being planned out ahead of time.
The thought is interesting because everyone recognizes that "friend" covers an immense spectrum yet we have few explicit demarcations. "To grow closer to someone" is commonly used and the closest match to your concept I can think of. Another very common way of expressing this experience is something like "we've been spending a lot of time together"/"I've been hanging out with X more"/"I've been getting to know X better" especially followed by a positive statement about the person or relationship ("and I can talk to him about stuff I actually care about"/"and we get along really well"/"and she's pretty cool").
But it is sort of roundabout. Honestly your description, "becoming better friends", sounds good as is and I'm sorry to say I don't think we have anything fancier.
Referring to time ("old friend"/"family friend"/"childhood friend") can be used to imply the closest kind of friend because a friendship that has withstood time has presumably lasted for a reason, plus you've had the time to get to know each other and strengthen the relationship.
Relationships can be "strengthened", although almost always in past tense, rather than describing an ongoing, gradual change (one rarely hears "our friendship is growing stronger"). It's most often used when overcoming some difficulty raised the relationship up a level. Perhaps your mom was sick and X "was there for you", helping you during that time; or someone spread a mean rumor and X didn't join in the gossip; or you hadn't spoken to X in a while but when you reunited, it didn't feel as if time had distanced you. When this happened "it strengthened our friendship". (To quote J K Rowling: "There are some things you can't share without ending up liking each other, and knocking out a twelve-foot mountain troll is one of them.")
You can say Things got Real:
When a situation changes from the tame and quotidian to the unfamiliar and hazardous, things got real. Things getting real can lead to unexpected and potentially life-threatening consequences. If you believe things may soon become real, the best thing to do is to get real yourself by staying calm and maintaining an awareness of your surroundings.
In a sense, imagine a platonic relationship where one of the parties constantly wished it was more. In a sense, that person would be living somewhat of a lie, pretending to be happy with the friendship as is. Once it goes to the next level though, then it just got real.
Yeah, technically this is more than a single word, but when you say 'Our relationship just got real', it flows.