At work, I have been using a solder station from Weller. It had a 80W iron, which was still very handy: Not longer that 18cm, a not too thick handle, and the heater was as thick as a pencil. The tip was exchangable, we had fine, needle-like tips for soldering SMDs as well as more bold ones for thicker cables like 2.5mm² and more.
Sometimes, I also soldered more heavy things like ground straps with it, but this isn't fun. The iron had not enough power, and due to it's size, also not enough heat capacity. I had to set the iron to 480°C and wait a moment before I could solder. The result was... well, there was an electrical connection, but I wouldn't call it reliable.
It is clear that this temperature is far to high, the flux just burned, the tin had a bad consistency, the tip didn't like it, and you can apply too much heat.
We also once had do solder lots of thick cables like 16mm², therefore we got a very strong solder iron, I guess a minimum of 150W.
So, my personal conclusion is that a regulated solder station in the range of 60-100W with exchangeable tips of different size is fine for standard jobs from small SMD soldering up to standard mains cables. For that 16mm², you should have an extra iron, I recommend at least 150W, or more. (Also, that 16mm² stuff is usually not that sensitive to over-temperature.)
P = EI, so I = P/E = 48/230 = 0.2 Amp.
Since this iron has some kind of temperature control, the incoming AC power will not be connected directly to the iron.
Best Answer
You don't need a $1000 iron unless you're going to be using it all day, every day, for a living, for the next 40 years. Same as any other tool. $100 will get you a good iron that will last you a long time (if you take care of it) and will do any hobby-level job.
Adjustable temperature is nice for versatility, but not necessary. You can overcome a fixed-temperature iron's limitations with technique and practice.
Personally I spent the first few years going through inexpensive (sub-$40) irons before finally setting aside $100 for a Hakko station. Things I noticed:
You can, of course, get by with inexpensive tools depending on your level of use and the types of projects you're doing. Many projects are absolutely doable with a $20 35-watt iron. To me, $100 seems just about right for a tool I use reasonably often at a hobby level.
Edit per recommendations from comments:
I submit that 35-40W is ample power for small projects that only involve small-gauge wire and component leads. I've used a 35W iron for things like swapping guitar pickups and little circuits you assemble on perfboard with 20AWG wire. More wattage is generally not a bad thing, as you'll generally end up with an adjustable-temp station above probably 60W or so (my Hakko is 70W). For soldering to a big hunk of metal (like a large grounding plate or block), you may eventually need a 100-150W gun.
I certainly haven't taken a survey of every available option, nor used them all, so as always YMMV.