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I can't seem to find the right angle to approach this concept intuitively. Does anyone have any ideas of how to consider hyper-operations in a way that isn't recursive, such as to accept non-integers?
I am not sure I get your problem correctly.

Take the function $f: b^z$ as to be iterated, with, say $b=sqrt(2)$ .
Assume one plane on a math-paper and look for easiness only the lines and their crossings of the coordinate-system of integer complex numbers $z_0$ .
Now take another paper, position it 10 cm above and for every point of the crossings (and ideally also of the lines) mark the values of $z_1=b^{z_0}$. Then repeat it with a third plane, again 10 cm above, marking $z_2=b ^{b^{z_0}}$ .
After that, try to connect the related points of the zero'th, the first and the second plane by a weak string, say a spaghetti or so. Surely except of the fixpoints in $z_0$ it shall be difficult to make a meaningful and smooth curve - and in principle it seems arbitrary, except at the fixpoints, where we simple stitch a straight stick through the iterates of the $z_0$ at the fixpoint.
Of course the spaghetti on the second level is then no more arbitrary but must be - point for point - be computed by one iteration. But the spaghatti in the first level follow that vertically orientated curve, where a fictive/imaginative plane of paper is at fractional heights and the fractional iterates would be the marks on the coordinate-papers at the "fractional (iteration) height".

I'd liked to construct some physical example, showing alternative paths upwards between the fixed basic planes, with matrial curves made by an 3-D-printer, but I've not yet started to initialize the required data.

But I think, that mind-model alone makes it possibly already sufficiently intuitive for you.

A somewhat better illustration is in my answer at MSE, see https://math.stackexchange.com/a/451755/1714
I think the OP refers to concepts like , what i called " semi- super " operators.

Like the semisuper operator of the semisuper operator of f(x) is the super of f(x).

This is extremely difficult.

Do not confuse with the functional half-iterate of the superfunction.

Regards

Tommy1729