I have plans for something like that in the future, but first I'd like to perfect the client itself (a persistent connection would require a server).

For now you can always try out https://www.irccloud.com or https://github.com/thedjpetersen/subway

https://github.com/leedo/alice is another one. Used to be hosted but they stopped running it.

During my time inside Trolltech/Nokia, everyone was on Freenode and our internal IRC server. A lot of people were connected through a client called Quassel [0] which is basically a distributed IRC client you can access from multiple machines (-> you'd host it on a server, it stores the backlog and everything else for you).

In my current startup we have the same (internal IRC server) and we also thought offering Quassel hosting [1] is a good idea.

[0] http://www.quassel-irc.org/

[1] http://quassel.woboq.com/

Shameless plug here as well, I'm developing something that will sort of solve that problem for channels you own: http://www.getinstabot.com

There have been irc bots for as long as there has been IRC.

Eggdrop bot is a common one, open source, etc.

> I'm connected and happily chatting, but then my Wifi blips and I'm disconnected. There's no chat log, so I don't know what I missed. And on crappy connections, this can happen pretty frequently.

So a bouncer + web interface, huh? I'd better get to work..