Does the string "...CATCAT..." appear in the DNA of Felis catus?

While Matt's answer is perfectly correct, it is important to note that the sequence $(CAT)_n$ in DNA is not restricted to cats, and you would expect to find it anywhere.

For example, searching the human genome for the same 3-tandem repeat CAT sequence results in many hits as well.

This is because you are essentially searching for short tandem repeats on the DNA strand. These repeats can occur in any organism, and therefore while finding CAT substrings in the DNA of the cat may be amusing, they aren't special to cats (or any other animal) and are only the result of an artifact of naming of the bases coincidentally matching with the name of the animal.

To augment the other answers, let's compute the probability of CATCATCATCAT occurring in random DNA sequence.

Cat DNA length is 2.7 gigabases (source), and there are 4 possible bases. For 1 CAT there are 3 bases, giving expected number of occurrences in 2.7 Gb as $frac{2.7 cdot 10^9}{4^3} approx 42,188,000$

Repeating the calculation for longer sequences gives:

• 1 CAT: 42 188 000 occurrences
• 2 CAT:      659 180 occurrences
• 3 CAT:        10 300 occurrences
• 4 CAT:             160 occurrences
• 5 CAT:                2 occurrences
• 6 CAT:                0 occurrences

So, indeed, there are many more CATs in cats than could be expected by pure chance alone.

So, there are a few great answers here already, but it seems nobody addressed an interesting part of your question: GEB was published in 1978 and the genome of Felis catus was not sequenced until many years later... so how did he know?

jpa's answer shows that you'd expect to get only about five CATs - not ten, and the chance of getting ten is astronomically low. I expanded his table to show the depressingly low chance of getting ten by perfect randomness:

5 CAT: 2.5 expected per Felis catus genome
6 CAT: 0.04 expected
7 CAT: 0.00061
8 CAT: 9.54 e-6
9 CAT: 1.49 e-7
10 CAT: 2.32 e-9

That means you'd expect to find 10 CATs about 0.00000000232 times per random genome. So how on earth did the Felis catus genome end up with ten CATs in it? And how did Hofstadter know that there would be this many CATs?

As it turns out, this repeated sequence of a few base pairs is called a "short tandem repeat", or "microsatellite". This is when a 2-5 base pair sequence is repeated several times, usually between 5 to 50 times.

So at this point, to recap: we know the chance of getting this 10 CAT sequence is slightly more probable, but since we're restricted to just the Felix catus genome we definitely aren't guaranteed a 10xCAT sequence. So how did Hofstadter state it as if it was a fact?

As it turns out, one critical property of STRs, or short tandem repeats, is that mutations in these areas are far more common, and they represent a large amount of the genetic variation between individual members of a species. This discovery was made with the advent of DNA sequencing, which began only a few years before the book was published. Therefore, given a large population of nonidentical cats (which we have), we can confidently say that there is an extremely high chance for a 10xCAT sequence.

Hofstadter's genius perfectly combined math (only 2.32e-9 expected sequences per genome) with biology (microsatellites increase the chance of finding this sequence) with forensic genetics (in a population of the same species, individuals are likely to have many STR-related differences.) All of this put together gave Hofstadter what he needed to confidently say: yes, CATCATCATCATCATCATCATCATCATCAT almost certainly exists in the Felis catus DNA. Little things like this are why Godel, Escher, Bach is my favorite book of all time.