A web IRC client whose connection belongs to you rather than to a browser tab. Shut the laptop and you stay joined; open it again and the conversation you missed is still there, read from disk rather than from a server's goodwill.
A bouncer is built in. Your networks stay connected with every window closed, and reconnect on their own after a restart.
Every line is written down, searchable across networks you have since left — including the ones you were never online for.
Install it to the home screen and a mention finds you through Web Push, with nothing open and no app store involved.
This is a small private server, so accounts are handed out by a person rather than a form. Tell me who you are and why you want one; you will hear back.