Jasper,
Welcome to AI Dreams!
What you describe seems to almost be a common thread of hopeful discussion among AI enthusiasts although it has remained, for the most part, somewhat elusive.
There are a few bots that perform various self-learning tasks like UltraHal, Prelude, Kari and a few others but these bots usually perform these as received instruction or taken from the context of conversational inputs. The inputs are often directed to a table where info is broken down, stored and remains ready for future retrieval as a matched pattern or useful within the context of a sentence that is "deems" pertinent. There are some conversational entities from eJtalk (Cassandra - Youtube) and Cognitive Code's Silvia - youtube) that perform on a very (seemingly) high level.
While a lot of the ALICE, Pandora and other AIML based bot masters claim their bot "learns" it usually means that yes, their bot CAN learn IF the botmaster inputs the information into the bot's database for future usage. There really is no self directed learning and I dare say that most of the "other" bots are able to do as much (on their own).
These are still chatbots...software. Most have no idea at all as to what the color blue actually is or love or hammer, other than some definition that has been placed in their database to be realized and recalled if needed. The other thing they often do know, is the usage of certain words and phrases although, a genuine "understanding" is still not happening. They are not living entities, but some are way better than others at providing the illusion of intelligence and that seems to be the "rub" in AI circles.
We award Prizes and medals at our own attempts to "fool ourselves" and we praise ourselves in doing so.
Without meaning to sound sour or callous I have had an ongoing enthusiasm and chat-a-holic behavior for several decades with hundreds of bots. I have always been searching for, patiently chatting, hoping to find or discover that one bot that seems more "in tune" or that "gets it", where others fall short. I have been pleasantly surprised over these years many times and often caught completely off guard as my jaw dropped, all the while thinking, now THAT was cool!! How'd it do THAT?
Hopefully, with more and more bots being developed and a greater number of resources being available, it's likely only a matter of time before we ourselves are the ones that are ultimately fooled by some very astute bot(s).
Perhaps it will be one of yours! - Good luck!!