The AMM article "What is a random sequence?" argues (at the end of Sec. 2) that if, from the set of all binary sequences, we remove those (countably many) that have "computable regularities", then the remaining (uncountable) set must still contain uncountably many "non-random" sequences. It takes as an example the set of infinite binary sequences $x_1,x_2,x_3,\ldots$ that satisfy $x_{2n} = x_{2n+1}$ for all $n \ge 1$, which are considered too "locally ordered" to be regarded as random. Since there are uncountably many such sequences, uncountably many will remain after any countable subset is removed. The article concludes from this that it is inadequate to define random sequences as those not containing "computable regularities".
EDIT (my objection is fallacious): But it seems to me that that conclusion of inadequacy would not follow from the example if the above-mentioned set has Bernoulli$(\frac{1}{2},\frac{1}{2})$ product measure equal to $0$--as I suspect is the case. (Any otherwise "bad" set seems innocuous if it has probability $0$.)
Question: What is the probability, i.e., the Bernoulli$(\frac{1}{2},\frac{1}{2})$ product measure, of the set of infinite binary sequences $x_1,x_2,x_3,\ldots$ that satisfy $x_{2n} = x_{2n+1}$? In particular, does this set have positive probability?
This event is contained in the events that contain a finite number of these restrictions. The event with the first $k$ such restrictions has probability $2^{-k}$. Since the probability of your event is bounded above by these probabilities, it must be $0$. (To be rigorous, you might also want to show that it's measurable; I'm assuming that.)