In the UK $100/year gets you about a 20-40w constant load*.
That's a fairly efficient machine, and then there's the router, modem etc.
Granted some of that already exists, and a Raspberry Pi or similar would fit in that budget.
* quick maths: $100/365 gives 27 cents a day or approx 1c per hour - at $0.50/kWh you have a 25w constant load and at $0.25/kWh you have 50w.
This is true when thinking in centralised big-site terms but there are other ways to achieve a large reach, some of them already close to main-stream. Peertube [1] is an example of such, using a combination of Activitypub and peer to peer (here: bittorrent) to create a "web scale" platform without needing a lot of money. Does it work? A good question, Peertube itself certainly works fine but it would be an interesting test to see how far it actually scales.
That's not how algorithms work. Most of the things we do are nlogn complexity. There are lots of breakpoints where 10x as many users cost you 11x as much in hardware, and we pretend it's going to cost us 8x. Any time your estimates are off by 40% should trigger your root cause analysis protocol.
In particular we are lousy at statistics. We have to be right in the middle of the consequences of misreading them before we act. When you have 10 servers you might have to deal with a deployment race condition once a month. When you have 30 servers you might have to deal with one every week. That starts to materially affect your estimates when you have five of those sorts of things going on.