For low maintenance, the cheapest geared option is to use a hub-geared bike. Since the chain never shifts, it's heavier gauge and experiences less wear. It's also fitted tighter, so is very unlikely to drop off.
As upofadown mentions, belt drives reduce maintenance even further - I have this setup on a hub geared bike I use for long tours. Before, I'd clean and oil my chain every 3 or 4 days on tour, now I just need to change the oil in the hub yearly. However, you can get a similar effect more cheaply by using a sealed chain cover, as seen on most Dutch bikes (it's no accident that these are the most popular commuter bikes there)
I find cable disk brakes to be another good low-maintenance option (compared to eg canti brakes constantly needing toed in, and changing pads is easy).
You can go tubeless to prevent punctures, but retrofitting that to existing wheels is messy and more work than I've had in a couple of years worth of punctures. For tour/commute I just rely on tyres with kevlar protection (eg Schwalbe Marathon Plus). They cost a little more than standard tyres but you save on tubes in the long run.
All of this increases weight, but not a lot. If you're looking for low maintenance, you're not racing, and you can probably easily lose more weight by riding your bike than you can shed by changing parts.
I ride with just the usual tools - multitool, levers, pump, spare tube - but another option is using a folding bike. I also have a brompton and would often ride that on a city commute with no tools at all; if things go wrong (which they never did) my backup plan was always to fold it up and jump on public transport, fixing it later.
And a crank falling off is _very rare_ on all bikes.