Allow me to clarify what I mean by "intuitively". Most of us here can solve one layer "intuitively", and what this means to me is that I solve it simply by following target pieces around the cube and being able to visualise a few moves ahead. I am not consciously using any algorithms (although of course any intuitive solve could be described with algorithms), and simply go by feel. This is contrasted with, say, the last layer, where while I understand the consequences of algorithms I use, applying them is still a bit of a black box, and I am not able to naturally understand how the cube is changing during the algorithm. In other words, I could not solve the last layer "intuitively".
It's a bit of a vague term, especially when you consider that well-learnt algorithms become somewhat intuitive with time - but hopefully you all know what I mean.
An intuitive 3x3x3 cuber may well come to establish their own algorithms of sorts, perhaps - but think of it what I means as like doing an intuitive first-layer solve, but for the whole cube.
My question - is there anyone out there who would describe themselves as solving whole cubes by this method?
EDIT: Just to clarify - I don't mean someone who solved it once, without help. I mean someone who can solve it over and over again, but who does not knowingly use any algorithms.