I've always seen the canonical example for this to be $\sec x$. For this and more examples, see R. Stanley's excellent article on the subject, "Differentiably finite power series"

Oops, above I am referring to D-finite, power series, but you are referring to D-algebraic power series. It is proved in "A gap theorem for power series solutions of algebraic differential equations" by L. Lipshitz and L. Rubel that $$\sum_{n=0}^{\infty}x^{2^n}$$ is not D-algebraic.

Another function that was proven not to be D-algebraic is the Gamma function, and this fact is due to Holder.

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I've always seen the canonical example for this to be $\sec x$. For this and more examples, see R. Stanley's excellent article on the subject, "Differentiably finite power series"

Oops, above I am referring to D-finite, power series, but you are referring to D-algebraic power series. It is proved in "A gap theorem for power series solutions of algebraic differential equations" by L. Lipshitz and L. Rubel that $$\sum_{n=0}^{\infty}x^{2^n}$$ is not D-algebraic.

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I've always seen the canonical example for this to be $\sec x$. For this and more examples, see R. Stanley's excellent article on the subject, "Differentiably finite power series"