A CTL* formula EFG p is equivalent to mu-calculus formula Y.(<>Y | νZ.(<>Z & p)). In this formula, the alternation depths are ad(Y)=ad(Z)=1.
Is there a CTL* formula that translates into YνZ(...) with alternation depths 2,1 for Y,Z?
And a similar question:
Is there a CTL* formula that translates into νXYνZ(...) with alternation depths 3,2,1?
Answer: Y.νZ.[](Y | pZ), which corresponds to CTL formula AFAG p.
Long answer to 1st question
For this answer I use the def of alternation depth taken from Handbook on Model Checking, Chapter 26.
For instance, the formula Y.(<>Y | νZ.(<>(Zp))) has alternation depths ad(Y)=ad(Z)=1, so the AD of the formula is 1 as well. Note that this formula can also be written as Y.νZ.<>(Y | Zp) which has AD of 2. So the original question asks about the minimal possible depth.
Now consider a slightly different formula Y.νZ.[](Y | Zp) of AD=2. In contrast to the formula with <> operator, this formula cannot be rewritten into a formula of depth 1 (I believe so). Finally, this formula corresponds to the CTL* formula AXFG p. This answers the original question.
Remark. I mistakenly thought that Y.νZ.[](Y | pZ) could be rewritten into a formula of AD=1 just like for <>, which prompted the question.
Answer to 2nd question
νX.Y.νZ.[](sZ | Y | Xn), which corresponds to CTL* formula AX(GF!n -> GFs).