I have found another nonlinear example at sosmath.com here but it appears to contain mistakes.

Also you can play with the examples of power series solutions in Maple using dsolve with the series option.

Some further examples could probably be found in connection with the study of the Painleve property and the Painleve test of integrability for (systems of) ODEs, see e.g. the book Integrability and Nonintegrability of of dynamical systems (full text available at this URL) by Alain Goriely and references therein.

Another useful keyword to search for is formal solution instead of the power series solution, although the convergence will probably not be discussed in the context of formal solutions.

I have found another nonlinear example at sosmath.com here but it appears to contain mistakes.

Also you can play with the examples of power series solutions in Maple using dsolve with the series option.

Some further examples could probably be found in connection with the study of the Painleve property and the Painleve test of integrability for (systems of) ODEs, see e.g. the book Integrability and Nonintegrability of of dynamical systems (full text available at this URL) by Alain Goriely and references therein.

Another useful keyword to search for is formal solution instead of the power series solution, although the convergence will probably not be discussed in the context of formal solutions.