For f(z)=z^2-1, the arboreal Galois group always lies in a new group M∞ and equals M∞ exactly when [K(√-x0,√(1+x0),ζ8):K]=16, which is true for infinitely many x0 over Q.
Benedetto, Xander Faber, Benjamin Hutz, Jamie Juu l, and Yu Yasufuku, A large ar- boreal Galois representation for a cubic postcritically finite polynom ial, Res
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The arithmetic basilica: a quadratic PCF arboreal Galois group
For f(z)=z^2-1, the arboreal Galois group always lies in a new group M∞ and equals M∞ exactly when [K(√-x0,√(1+x0),ζ8):K]=16, which is true for infinitely many x0 over Q.