They shouldn't. Possible mistakes:
- transient: forgetting to throw away enough initial iterations - if you want to calculate the exponent associated with the attractor, you must wait until your trajectory converges to it;
distance rescaling between the trajectories: as the OP source (Sprott) reminds:
Readjust one orbit so its separation is $d_0$ and is in the same direction as $d_1$ [...] This is probably the most difficult and error-prone step.
Also, remember that the initial separation between the two orbits should be as small as numerically feasible, since the exponent is a local property.
average: an error that could explain the OP result of exponents that grow with the number of steps would be to calculate the sum of exponents instead of their average.
Of course, the list above isn't exhaustive.
More sophisticated methods exist, which don't require much more than a bit of matrix handling and simple derivatives. I'd highlight the one presented in Ott (1st ed., Eq. 4.40), but other textbooks on nonlinear dynamics and even
Wikipedia and Scholarpedia entries describe how to calculate the exponent.
And, last but not least, there is already an answered question on precisely the calculation of the Lyapunov exponent of the double pendulum:
Max Lyapunov Exponent of a Double Pendulum.