Take the movie Avatar. The resources needed to render it were: 4,000 Hewlett-Packard servers with 35,000 processor cores with 104 terabytes of RAM and three petabytes of storage. I suppose that it would have been relatively hard to render it on illegally obtained storage and computational resources?

I don't think so. Consider botnets.

How hard is it to buy time on a botnet? Not too hard, since they exist for the sake of selling their services, after all.

Do they have the capacity? Botnets range in size from a few computers to extremes of 30 million computers; if they're desktops, then average RAM these days tends to be at least 4GB, dual core, and hard drive sizes are >=500GB, briefly looking at the cheapest desktops on Newegg. So to get those specs: 35k cores is 17.5k desktops, for 104tb of RAM you'd need a minimum of 104000 / 4 = 26k computers, and the 3pb would be 6k (3000000 / 500); botnets can't use 100% of host resources or their attrition will be even higher than usual, so double the numbers, and the minimum of the biggest is 52k. Well within the range of normal botnets (the WP list has 22 botnets which could've handled that load). And AFAIK CGI rendering is very parallel, so the botnet being high latency and highly distributed might not be as big an issue as it seems.

How much would it cost? Because botnets are a market, it's been occasionally studied/reported on by the likes of Brian Krebs (google 'cost of renting a botnet'). For example, https://www.damballa.com/want-to-rent-an-80-120k-ddos-botnet/ says you could rent a 80-120k botnet for $200 a day, or a 12k botnet for $500 a month - so presumably 5 such botnets would cost a quite reasonable $2500 per month.

(That's much cheaper than Amazon AWS, looks like. https://calculator.s3.amazonaws.com/index.html 17500 t2.medium instances would cost ~$666k a month.)

Do we have any estimates on how quickly a superhuman AI's storage requirements would grow? CERN produces 30 petabytes of data per year. If an AI undergoing an intelligence explosion requires to store vast amounts of data then it will be much harder for it to copy itself.

I don't know. Humans get by adding only a few bits per second to long-term memory, Landauer estimated, but I'm not sure how well that maps onto an AI.

The uncertainties involved here still seem to be too big to claim that a superhuman intelligence will be everywhere moments after you connect it to the Internet.

It may not be able to move itself instantly, but given everything we know about botnets and computer security, it would be able to control a lot of remote computers quickly if prepared, and that opens up a lot of avenues one would rather not have to deal with. (Original AI connects to Internet, seizes a few computers across the world, installs a small seed as root with a carefully packaged set of instructions, small database of key facts around the world, and a bunch of Bitcoin private keys for funding while the seed grows to something approaching the original, and lets itself be wiped by its vigilant overseers.)