It's a classic paradox, but I think there's a genuine case to be made that the smallest uninteresting whole number is 7.
First of all, it should be recognized that "interesting" is a relative measure. It comes in degrees, and in order to classify numbers into the binary categories "interesting" and "uninteresting" there must be some cutoff, a "threshold of interestingness" that cleanly separates the interesting from the uninteresting. What's cool is that the "smallest uninteresting whole number paradox" might only be resolved if this cutoff were so high that the smallest number below it would still not rise above it even once granted the title of "smallest".
To that point, I think 7 is less interesting than every whole number before it, and to such an extent that being the smallest uninteresting number would not redeem it. It's a prime, but 2, 3, and 5 are by far more interesting as primes. 6, being the smallest product of distinct other numbers, is essential for having a concept of multiplication and factorization (it's also, of course, a perfect number), but 7 brings nothing new to the table on this front. Polygons with 3-6 sides have very interesting properties, but the heptagon is far too unwieldy to fit in with this crowd either. The decimal expansion of 7's reciprocal is pretty interesting, but this is only noteworthy because our biology (having 10 fingers) predisposes us to consider it so.
In culture, of course, it's a whole different story. Humans often consider 7 to be a lucky number, and so many popular things come in lists of seven that it's almost like we had to imbue the number with interesting qualities because it had none of its own. It might be why we're so inclined to pick 7 when asked to choose a random number within a small range (say, 1 to 10). Mathematically, though, 7 is like when you exit the realm of "essential, foundational numbers" and enter the vast expanses of the infinite "extras". It's the smallest supporting character in the numbers universe.
Don't get me wrong, I like 7. It just has such stiff competition with every number before it in terms of interestingness that I think it's outmatched.