A binary sequence $A=A(0)A(1)\ldots$ is called infinitely often (i.o.)~Turing-au\-to\-re\-duc\-ible if A~is reducible to itself via an oracle Turing machine that never queries its oracle at the current input, outputs either $A(x)$ or a don't-know symbol on any given input~x, and outputs $A(x)$ for infinitely many~x. If in addition the oracle Turing machine terminates on all inputs and oracles, A~is called i.o.~truth-table-autoreducible. We obtain the somewhat counterintuitive result that every Martin-L\"of random sequence, in fact even every rec-random or p-random sequence, is i.o.~truth-table-autoreducible. Furthermore, we investigate the question of how dense the set of guessed bits can be when i.o.~autoreducing a random sequence. We show that rec-random sequences are never i.o.~truth-table-autoreducible such that the set of guessed bits has positive constant density in the limit and that a similar assertion holds for Martin-L\"of random sequences and i.o.~Turing autoreducibility. On the other hand, we show that for any rational-valued computable function~r that goes nonascendingly to zero, any rec-random sequence is i.o.~truth-table-autoreducible such that on any prefix of length~m at least a fraction of~$r(m)$ of the m~bits in the prefix are guessed. We include a self-contained account of the hat problem, a puzzle that has received some attention outside of theoretical computer science. The hat problem asks for guessing bits of a finite sequence, thus illustrating the notion of i.o.~autoreducibility in a finite setting. The solution to the hat problem is then used as a module in the proofs of the positive results on i.o.~autoreducibility.