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Hmmm, seems like my wish list could get out of control. Would these items roughly match what you had in mind?

Fixed points


Convergence of Newton's method, interpolating polynomials
Kneser Riemann mapping, singularities
Leau-Fatou flower
Half iterate convergence, special case for sexp(z)-1, eta=exp(1/e)
Theta(z), implementation of Kneser algorithm, eta
Base change, Jay's and Sheldon's base change sexp, Nowhere analytic, Tommy's 2sinh iteration
intuitive iteration
Cauchy Method (Dimitrii Kouznetsov)

cheta function