That's pretty difficult, most large language models today have a set context limit which differs depending on the LLM that's being used.
The longer the conversation goes on, the more the LLM will forget.
One way to circumvent this would be to fine-tune the existing model with the information that you want to add, but this takes a lot of computing power and time. I've seen someone do this with a really small model which means that the fine-tuning process only took a few minutes, but the downside is that the model because of its small size was really "dumb" and hallucinated a lot (making up stuff basically).
The other way I've seen people do this is to have a database in which the information gets stored, but this also leads to frequent hallucinations, especially if the information listed in the DB and the one that comes from the LLM is conflicting.
So I'd say the best solution right now is just to use a model that has a large context out of the box.
I really don't know which kind of LLM the service that you used utilizes, if it's GPT4 then forget about any alternative since you probably won't get close right now.
I also can't really recommend any kind of online service since I really don't use them, but if you want to try a local model that runs on your own hardware then one model that's pretty popular right now is
mixtral which has a context of 32k tokens by default.
But it all depends on what kind of hardware you have.
Edit:
Apparently talkie uses
Claude as the LLM if that is true that would mean that it has a 200k token context limit but the way that they achieved that isn't optimal which means that the LLM tends to forget a lot even before the context limit is reached.