You can use a bouncer, like znc. This sits inbetween you and the IRC server and you connect to it when you want to connect to IRC, I would not recommend using a free one, as they can see all your passwords and log every channel you're in, but you could set it up on a server if you have one
An IRC bouncer is good for this purpose. It can always stays connected to your servers, then you setup your client to connect to it. They can be configured to "play back" what happened since you left each time you reconnect.
I run ZNC on a crappy Linux box and open a port through my router to it so I can take my laptop on the go with me.
There are also hosted bouncer services if you don't want to run your own.
He uses Quassel for both.
Quassel has the "client" and the "core" versions so that you can put the "core" on the droplet and connect as many clients as you want to on whatever device you want. - http://quassel-irc.org/
If you want a more universal IRC Bouncer then I would suggest ZNC. - http://wiki.znc.in/ZNC
Set umode +x on connect and connect via SSL, the standard port for which is 6697. For some servers you can log in to services via the server password which is sent on-connect, so if you have a vHost via HostServ or similar it will be activated immediately as opposed to whenever you identify manually.
If you have an external server or access to a shell account which allows IRC, you can set up a BNC. A BNC basically acts as a proxy for IRC, with the added benefit of being always online, so your home (or work, or school) IP address will be hidden and instead replaced with the IP address of your server. Most BNCs will save messages for you while you are offline and can be connected to via a regular IRC client. I use ZNC, try it out.
The bouncer is logged in as you. It acts as both a client that logs all activity (if you wish) and also an IRC server that you can connect to. Bouncers allow you to have a persistent connection even when mIRC is closed/your main computer is off, all from one nick. You can connect to it from any client you wish, even multiple at once. The logs get stored on your Pi. You can view them over SSH, copy them to your desktop with SCP, email them to yourself with a script, etc. You can also view them from within mIRC. That's actually why I use mine. I'm Op on a channel. My bouncer plays back everything that happened since I was last connected to it right I'm my chat buffer on WeeChat or my Android IRC client.
Edit for extra clarity: the bouncer needs to sit between you and the server, otherwise it couldn't capture your private messages. You can connect to the bouncer with any number of clients at any given time and the channel will only see your one nick. If you log directly into the IRC server (i.e. Freenode) then you'd have to have a second nick so they could tell you apart from the bouncer.
Another edit: link to ZNC.
Sounds like you need an IRC boucer (BNC). I have ZNC installed on a cheap VPS, works great. You set ZNC to connect to the IRC server, and connect your IRC client to your ZNC server. When you logoff on the client, ZNC stays connected to the server. When you reconnect to ZNC, it will play back missed messages.
Bought a few IPv4 NAT VPS' through http://lowendspirit.com/locations.html
IRC proxies and bouncers are nothing new. I've had ZNC running on my VPS for years. I can't possibly imagine choosing to install TapChat instead of ZNC after comparing their respective websites: the TapChat website gives you essentially no information about the software.
I'm fairly certain there was an /r/sydney IRC room for a while, but that was very very short lived.
As for people who never log out, they'd be using an IRC bouncer (I use this: http://wiki.znc.in/ZNC)
You could use a web client like qwebirc with a bouncer like znc to achieve what you're looking for, but would require you hosting it on a VPS or similar and you'd have to configure where should connect and what modules you want loaded.
Disclaimer: I only know of irccloud because of their presence on freenode, and I have never administered ZNC myself, although I have provided shells for others to do so. (I use irssi in screen and qwebirc exclusively.)
I use mine for a variety of silly things, but primarily as an IRC bouncer.
Basically my server connects to my IRC servers and channels of choice, and then I can connect to my server with as many clients (within reason) as I want and do everything from the same nick. Also, it logs everything.
But my true motive was to always win at IdleRPG.
Yea that sucks, you'll have to just re-queue if you DC or you could always setup an irc bouncer such as ZNC which will keep you online even if you disconnect. Mods are usually busy so while they might be online they are probably doing something else.
If you have access to an always-on server (such as an internet-connected computer at home), you can setup an IRC bouncer.
A bouncer sits between you and a given IRC server. The bouncer will keep you connected to one or more servers/channels, and will log messages while you are not connected to the bouncer. You can then connect to the bouncer from any device, see any messages that you missed, and continue to use IRC as normal.
I personally use ZNC.
If I look at http://wiki.znc.in/File:Webadmin-settings-dark-clouds.png from http://wiki.znc.in/ZNC I see that the servers listed in ZNC's network interface, allow for a password to be specified after ther server's host and port.
So quassel-irc is like ZNC but then you are forced to use their client?
If that is right, I would recommend ZNC, does about the same thing (being a IRC bouncer) and thus staying connected while you are gone and when you connect using your favorite IRC client to ZNC you get to see the previously received messages (as many as you like). Also allows you to connect from multiple locations (desktop/laptop/phone) at the same time while only having one nick in the channel.
It should also be possible to receive notifications when you are mentioned but not connected, but I have no experience on that part.
A bouncer basically works as both a client and a server. You need a place, such as a VPS, to run your bouncer from. It connects to the server as a client, and you connect to the bouncer as a server. It sits in the channels all day so you don't have to. You will be able to appear online and will be able to read a playblack of what happened in the channel after your return to the channel.
I use a bouncer called ZNC, the docs there are meh, but it's possible to get it working from. You will connect to the server via your bouncer as an IRC server.