I think there are two angles to this (as some have intimated):
-Most non-negative appliance (i.e. thing that stops you having to do difficult chores):
For me, like others, this has to be the washing machine. Although technically that is in the utility room. Dishwasher I think is a good investment, even going from a slimline to a fullsize has brought benefits in terms of not needing to put it on so often. I used to think them a bit of a luxury, but in terms of spend they work out at say £5/month plus running cost when averaged over a typical lifespan, and a month of not washing up is easily worth that. Also when working properly they get things a lot cleaner than washing by hand, glasses in particular.
-Most positive appliance (something that makes you smile but isn't as 'necessary' as the above, slightly more of a luxury, more about adding value than simply chore-dodging):
George Foreman grill. Makes grilling easier, can use alongside the oven, a lot better than our old Breville grill (plus added bonus of dishwasherable grill plates!)
I dunno, when you think about it most of the 'standard' items people have in kitchens all have their place in modern society and have done for years.... hard to imagine life without a cooker for example.
Would be interesting to hear what 'fantasy' items people have, i.e. appliances they'd like but don't yet have, for me I'd say maybe a massive great ****-off fridge freezer or seperate units so I don't have to faff about cramming stuff in.
Likewise what are your most under-used appliances:
-Tumbledryer: Just been sat in a corner taking up space for over 4 years since we moved in, never switched on I think
-Microwave: I use it maybe once a month