The maximum length of a string of consecutive primes is 2: that is, the primes 2, 3. This is easily proven, as no even number other than 2 is prime.
In contrast, consider the set of numbers which are a product of exactly two primes (they don't need to be distinct). This set begins 4, 6, 9, 10, 12 ... It goes on to include the numbers 33, 34, 35 - a run of three consecutive integers. Is this the longest consecutive run of such numbers?
My conjecture: that each set of numbers which are the product of exactly n primes contains one and only one run of consecutive numbers which is n+1 long.
No idea whether this is true! Thoughts, anyone?