Aphid
Regular.
I've manage for just over a year without a brew fridge, instead using a trug and aquarium heater for temperature control. The downside is that I can't control the temperature in the summer, and saw a drop in the quality of my beer because of it, but it works perfectly in winter when you know the ambient temperature won't go above fermentation temp. Will be moving house next year, and brew-fridge space will be an essential requirement.
In terms of absolute essential: A bucket, siphon, sediment cap, bottling wand, cheap capper, thermometer, hydrometer, some bottles and a way to control temperature for kit brewing. For all grain you will also need a boiler and a grain bag. In terms of consumable you need steriliser, bottle caps and ingredients. That is all you need to brew great beer. Don't get me wrong - I love having a chiller, spare fv to rack to with tap and little bottler, star-san, oxi cleaner, trial jar, bench-capper, hop strainer, large stock pot for sparge water etc..., but I only have them because I want them and they make my life easier.
Thanks for the comprehensive list. Very helpful. YouTube videos show the initial grain and hops being boiled inside muslin bags within a large stainless steel pot. As good as a boiler? The Starter Kit I have in mind comes with a barrel rather than bottles. Isn't this an easier way to start?