TITLE: Are these continued fractions of integrals known?
QUESTION [16 upvotes]: Motivated by this paper on polynomial continued fractions (Bowman, 2000), I thought about various extensions to current definitions of these fractions.

What if we defined a continued fraction such that the numerator of each fraction is the integral of the previous one?
That is, define the generalised integral continued fraction (GICF) as \begin{align}\mathcal G(f(x))=\dfrac{f(x)}{1+\dfrac{\int_0^x f(t)\,dt}{1+\dfrac{\int_0^x\int_0^t f(u)\,du\,dt}{1+\cdots}}}=\dfrac{f(x)}{1+\dfrac{f^{(-1)}(x)}{1+\dfrac{f^{(-2)}(x)}{1+\cdots}}}={\large{\mathop{\mathrm{K}}_{n=0}^{\infty}}}\frac{f^{(-n)}(x)}1\tag1\end{align} where $f(x)$ is a continuous and infinitely differentiable function on $\Bbb R$, and $f^{(-k)}(x)$ is the $k$th antiderivative of $f(x)$ using Lagrange's notation.

 Clearly $\mathcal G(0)=0$. 
There are indeed many special cases. For example, the derivatives of $\sin x$ are eventually periodic (of order $4$) and graphing software reveals that $\mathcal G(\sin x)$ is not convergent, since $${\large{\mathop{\mathrm{K}}_{n=0}^{4r}}}\frac{\sin^{(-n)}(x)}1=\begin{align}\begin{cases}\sin x,\quad&\text{for}\,r\,\text{even}\\-1-\cos x,\quad&\text{for}\,r\,\text{odd}\end{cases}\end{align}\tag2.$$ Now we move on to the simplest non-trivial case: when $f(x)=1$. Of course, we have that $$f^{(-k)}(x)=\frac{x^k}{k!},\tag3$$ the $k$th term of the Taylor expansion of $\exp x$. Then $$\mathcal G(1)=\dfrac{1}{1+\dfrac{x}{1+\dfrac{x^2/2!}{1+\cdots}}}\tag4$$ and plots for approximating $\mathcal G(1)$ are below (from $n=15$ to $20$). 
We can see that as $n$ increases, $\mathcal G(1)$ contains more 'waves' - and clearly $\lim\limits_{x\to-\infty}\mathcal G(1)$ does not exist since it is zero for odd $n$ and $+1$ for even $n$ (thanks @PeterForeman). However, I believe there are more things that can be investigated.

Questions

Has the GICF already been studied/discovered? If so, reference to literature would be appreciated.

With regards to the $\mathcal G(1)$ function, is it true that the maximum value of each 'wave' increases as $x$ increases up to the first wave? Are there an infinite number of such waves, and is there a closed form for $\max\mathcal G(1)$?

REPLY [2 votes]: Not an answer, but one more picture, with the functions for 100 and 150 terms superposed.

It is not clear what is happening on the left.
EDIT:
Here is the code (using sagemath):
def fun(x, N=150):
    L = [x**k/gamma(k + 1) for k in range(N, -1, -1)]
    resu = 1.0
    for t in L:
    resu = t / (1 + resu)
    return resu
P1 = plot(lambda x:fun(x, 100),-12, 3, aspect_ratio=1, color='red'); P1