A pressing need for processing data post-collapse is weather forecasting. Knowing the upcoming weather can help avoid crop failures from planting at the wrong time. Of course the data aggregation would also be very challenging as you need data from remote sites for good forecasting.
It might also be nice to know where (and when) groundwater is safe to drink
our descendents will perhaps not be thrilled to learn about the "time bombs" we have left them, steadily inching into aquifers or up into surface water. that is of course if they are not too distracted locating water of even dubious potability to care
Weather forecasting before computers already relied on rapid telecommunications of low-bandwidth digital data (temperature, pressure, humidity, wind speed and direction, and precipitation) from a network of weather stations. Digital telecommunications is something that computers and radios can provide at enormously lower cost than networks of telegraph cables. See https://news.ycombinator.com/item?id=43484415 for details.
Even a little data from remote sites can provide a huge advantage for forecasting. Temperature, humidity, and air pressure, roughly three bytes, four times a day: 0.001 bps per weather station. (Precipitation and wind speed and direction are pretty useful, but worse cost-benefit.) And collection of that data is very much less labor-intensive when a microcontroller and radio does it for you.
Other kinds of very-low-bit-rate telecommunications messages that are still extremely valuable:
"Lucretia gravely ill. Hurry."
"I-44 mile 451: bandits."
"Corn $55 at Salem."
"Trump died."
"Springfield captured."
"General Taylor signed ceasefire."
"Livingstone found alive."
The first of these inspired Morse to invent the telegraph; she died before the mail reached him. None of them are over 500 bits even in ASCII, and probably each could be encoded in under 100 bits with some attention to coding, some much less. 100 bits over, say, 2 hours, requires a channel capacity of 0.014 bits per second.
Even without advanced compression algorithms, you could easily imagime the corn message being, say, "<figs>!$05000<ltrs>ZCSXV" in ITA2 "Baudot" code: 14 5-bit characters, 70 bits.
Information theory shows that there's no such thing as being out of communication range; it's just a question of what the bit rate of the channel is. But reducing that to practice requires digital signal processing, which is many orders of magnitude more difficult if you are doing it with pencil and paper. It also benefits greatly from precise timekeeping, which quartz resonator crystals make cheap, reliable, robust, and lightweight.
Encryption is another case where an amount of computation that is small for a microcontroller can be very valuable, even if you have to transmit the encrypted message by carving it into wood with your stone knife.
The Bitcoin blockchain in its current form requires higher bandwidth than a weather station network, but still a tiny amount by current internet standards, about 12kbps originally, I think about 26kbps with segwit. Bitcoin (or an alternative with a longer block time) could potentially provide a way to transmit not just prices but actual payments under adverse circumstances. It does require that participants have enough computing power to sign transactions; I think it should be relatively resilient against imbalances of computation power among participants, as long as no 51% attack becomes feasible through collusion.
I wrote, "<figs>!$05000<ltrs>ZCSXV". This should have read, "<figs>!$05500<ltrs>ZCSXV".
Also, see https://news.ycombinator.com/item?id=43487785 for a list of other end-uses for which even a microcontroller would provide an enormous advantage over no electronics at all.
Bitcoin will be useless in that case. Half of the techbros wouldn't even survive the early 90's. In the 80's, forget something like properly learning Forth with Starting Forth and doing something useful.
Bitcoin already withstood a rapid withdrawal of more than half of the mining power over about a month, that time the PRC outlawed Bitcoin mining. And it also survived the relatively sudden collapse of Mt. Gox, which accounted for significantly more than half the trading at the time. And it survived its price collapsing by more than half in 24 hours, from over US$8000 to under US$4000. It seems a have pretty good survival characteristics.
In an environment where there isn't a world hegemon to run something like the post-Bretton-Woods system, international payments, if they are to happen at all, need to be settled somehow. The approach used for the 3000 years up to and including the Bretton Woods period was shipping gold and silver across oceans in boats. Before that, the Mediterranean international economy was apparently a gift economy, while intranational trade in Mesopotamia used clay bills of deposit.
In a hypothetical post-collapse future without a Singularity, there may be much less international trade. But I hope it's obvious that international trade covers a spectrum from slightly advantageous to overwhelmingly advantageous, so it is unlikely to disappear altogether. And Bitcoin has overwhelming advantages over ocean shipping of precious metals. For example, it can't be stolen in transit or lost in a shipwreck, and the latency of a payment is about half an hour rather than six weeks.
And all the blockchain requires to stay alive is about 26 kilobits per second of bisection bandwidth.
