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How big is the blockchain going to be in another ten years?
At its current rate with no new growth for 10 years, I would estimate around 382 Gigabytes.
If growth doubled each year for 10 years, I would estimate around 65.5 Terabytes.
(Edit: as pointed out below, my 65TB estimate does not consider the blocksize limit)
How big is the blockchain going to be when the last satoshi is mined?
There is no point in estimating since by that point in time, we would have already implemented some form of "pruning"
or other system that would help with the possible excessive file size. It could be as high as 680 trillion yottabytes.
But I highly doubt that since we would have taken steps by the time the size is around or before 1 Terabyte.
Can something be done about this? Maybe having parts of the blockchain archived?...
Yes. This is a known issue and there have been discussions about this with different possibilities as answers.
One was placing "checkpoints" in the blockchain where we hash all prior txs/blocks and agree that hash is absolute,
and then find the next block from that hash (instead of the prior block found) like a new genesis block.
That would effectively be like "the blockchain archived". This idea, like all others, has pros and cons.
Who is going to run a node if it takes a month to synchronize?
Yes. This is a known concern to the decentralized nature of Bitcoin/bitcoin.
It is important to note, that Satoshi said himself that a day would come in the future where average users would
no longer reasonably be able to maintain the full blocksize and data centers dedicated to this purpose many need
to exist and be used. When that will be depends on how the data is handled now and in the future.