How to verify the order of DOPRI Runge-Kutta method?

You're just hitting machine precision. Verify the slope of the log for error values larger than say $10^{-12}$ as for smaller errors, you'll see rounding errors due to the double precision arithmetic of your computer, and to the finite number of digits of the coefficients of the butcher table you have (quite often in textbooks, you don't even have the first 16 digits).

If I remember correctly, in the special case where $f(t,y)$ is a polynomial in $t$ of degree $d$ and independent of $y$, every Runge-Kutta method of order $d+1$ gives the exact solution (which is a polynomial of degree $d+1$). So the Euler method solves $y' = 1$ exactly and the classical fourth-order RK-method solves $y' = t^3$ exactly (all up to round-off error).