So I was mixing smooth functions with fractional calculus when I came upon the following idea. We could have functions in $D^\alpha$ defined as follows:
$f(x)\in D^1\iff\frac d{dx}f(x)=f'(x)$ exists.
$f(x)\in D^2\iff\frac{d^2}{dx^2}f(x)=f''(x)$ exists.
$f(x)\in D^k\iff\frac{d^k}{dx^k}f(x)=f^{(k)}(x)$ exists.
$f(x)\in D^\omega\iff\forall k<\omega(f(x)\in D^k)$, which is equivalent to $f(x)$ being a smooth function.
Next, we need to define fractional derivatives. Per this question, I don't mind which fractional derivative you use, so long as it is one of the many well known and accepted fractional derivatives.
Now, we may extend our derivatives further to what I will call hyper smooth functions.
$f(x)\in D^{\omega+1}\iff\frac d{dt}f^{(t)}(x)=f^{(1,t)}(x)$ exists.
$f(x)\in D^{\omega+2}\iff\frac{d^2}{dt^2}f^{(t)}(x)=f^{(2,t)}(x)$ exists.
$f(x)\in D^{\omega+k}\iff\frac{d^k}{dt^k}f^{(t)}(x)=f^{(k,t)}(x)$ exists.
$f(x)\in D^{\omega2}\iff\forall\alpha<\omega2(f(x)\in D^\alpha)$
$f(x)\in D^{\omega2+k}\iff\frac{d^k}{du^k}f^{(u,t)}(x)=f^{(k,u,t)}(x)$ exists.
$f(x)\in D^{\omega3}\iff\forall\alpha<\omega3(f(x)\in D^\alpha)$
And so on, defining ourselves $D^\alpha$ for every $\alpha\le\omega^2$. A function is called ultra smooth if we have:
$f(x)\in D^{\omega^2}\iff\forall\alpha<\omega^2(f(x)\in D^\alpha)$
But are these functions unique? Is $\{f(x)\in D^\omega\land f(x)\notin D^{\omega+1}\}$ an empty set or not? It seems rather hard to find examples (and don't forget you are free to use whichever fractional derivative you choose). In general, can a function be in $D^\alpha$ but not in $D^\beta$ for $\omega\le\alpha<\beta\le\omega^2$?
Assuming we enforce a domain restriction that everything must be $\mathbb R\mapsto\mathbb R$, exponential functions are an appropriate example. Under certain definitions and $a>0$, we have
$$\frac{d^t}{dx^t}a^x=a^x(\ln(a))^t$$
However, when $0<a<1$, we find that $\frac d{dt}\frac{d^t}{dx^t}a^x$ does not exist (as a function $\mathbb R\mapsto\mathbb R$) and so this is an example of a function that is smooth but not hyper smooth. Likewise this pattern can be used to show the existence of functions $f(x)\in D^{\omega k}$ and $f(x)\notin D^{\omega k+1}$.
However, I cannot find any examples where $f(x)\in D^{\alpha+1}$ but $f(x)\notin D^{\alpha+2}$.
Allowing functions to be $\mathbb C\mapsto\mathbb C$, I cannot find any other examples of this.