Everybody is going to have a different take on this. So here's mine!
I had a turkey fryer setup borrowed from my neighbor, and I've mostly replaced it with my own gear since I started a year and a half ago.
My brew process goes something like this: Measure my strike water and start heating it, mill the grain, combine in the cooler and start the mash, measure my sparge water, heat that and transfer to the HLT (or just heat it in a separate pot), drain the mash and batch sparge it, boil the wort, chill the wort, drain to the bucket, rehydrate my yeast, pitch it, wait 3 weeks, bottle it, wait 2 weeks, drink it. There are plenty of science-y steps along the way like checking gravity and temperature and cleaning the gear, but those aren't actually required to make beer.
My main equipment is this:
- Propane burner and tank - can't do much without this!
- 15 gallon aluminum pot - I got it at a kitchen supply store for $60 and added a ball valve from BargainFittings.com. It's fantastic.
- 54 quart picnic cooler - I added a CPVC manifold instead of the toilet braid.
- Copper coil immersion chiller - You could let your brew chill overnight with no help (I know a LHBS owner who does just that), but if you want to get it done quickly you need a chiller of some sort.
- 8 gallon fermenting/bottling buckets.
- Thermometer, hydrometer, big spoon, etc.
I also have a 5 gallon water cooler I use as a hot liquor tank, but recently I've just been using one of the turkey fryer pots I still have on hand so I can keep my brew kettle available for collecting wort. I make use of my neighbor's grain mill to double-crush the grains I get from the LHBS, but that's not necessarily required, I just feel like I get better efficiency, and it's there so I use it. And until I throw out my back lifting the mash tun, HLT, or full buckets up to my workbench, I don't have any need for pumps.
When I think of brewing essentials, I think big burner, brew kettle, cooler (mash tun), and a HLT so I can free up my brew kettle when it's time to collect the mash runnings. Boil it, chill it (or wait), drain it into the bucket, pitch the yeast and let it ferment. I also use an auto-siphon that I truly hate and am about to ditch, and I just started using a Johnson temp controller to keep my brew from getting too cold in the garage. Also, one of the best mods I've made to my brew setup is adding a hose bib to the incoming water line on my water heater. I connect an RV drinking water hose to that and I use it to collect my brew water and to run through my immersion chiller. It's as direct from the ground water as you can get, and it's got great flow! So much easier than lugging water from the kitchen sink or running hoses from the side of the house.
What'd I miss?