I used to have a 030 accelerator and enjoyed that a lot before I sold it.
Right now I have an 8 mb fast RAM expansion, a CF card with an IDE adapter, a SD card PCMCIA adapter and a 3c589 network card, and it definitely gets me by. I can move files from a more modern computer using the SD card adapter or download directly from aminet and so on using the network card. The extra RAM is more than enough for me in most cases but I admittedly mostly play whdload patched games and use Protracker with the thing.
I have the network card directly connected to a Raspberry Pi to be able to use the wireless network, and on the Pi I also run a gopher server that I use for file sharing.
The "minimum" is only a matter of what you want to do with the thing, but I'd still recommend the RAM expansion as it gets you quite far, and I don't really miss the accelerator that much. Maybe it's useful if you are into old flight simulators or the type of games that run slower on a stock machine. As for LCD monitors, I have a cheap kitchen TV with RGB scart input that I use some times, but in the end I still prefer the 1084S.