I. Quintic. The general quintic can be reduced to the form,
$$x^5=p+x\tag1$$
$$x = \sqrt[5]{p+x}$$
Hence by an iterative process,
$$x =\sqrt[5]{p+\sqrt[5]{p+\sqrt[5]{p+\sqrt[5]{p+x\dots}}}}$$
Truncating yields $x$ in terms of $p$ to any desired degree of accuracy.
II. Sextic. Likewise, the general sextic can be reduced to,
$$x^6 = x^2+2px+q\tag2$$
Or equivalently, let $D=-p^2+q$,
$$x^6 = D+(p+x)^2$$
$$x =\sqrt[6]{D+(p+x)^2}$$
By a similar iterative process,
$$x =\sqrt[6]{D+\left(p+\sqrt[6]{D+\left(p+\sqrt[6]{D+\left(p+x\dots\right)^2}\right)^2}\right)^2}$$
and so on, it can be seen that the roots of the general sextic can be expressed as an infinitely nested radical as described (albeit less concisely) in this pre-print by Nikos Bagis.
III. Septic. The general septic can be reduced to the form,
$$x^7 = x^3+px^2+qx+r\tag3$$
Bagis claims that an analogous iterative process will show that the roots of the general septic is also an infinitely nested radical. However, his eqns $22-33$ are unconvincing (especially $22-24$). An obvious way is to express $(3)$ as,
$$x^7 = (x+\alpha)^3+\beta$$
This gives us only two unknowns $\alpha,\beta$, but there are three variables $p,q,r$.
Question: Are the roots of the general septic really an infinitely nested radical analogous to the quintic and sextic cases?
P.S. This was inspired by this post.