Use spectral methods? But you’d presumably also have to use higher precision number types if you want that many digits.

Of course you'd have to use arbitrary-precision arithmetic, that's the easy part. The hard part is knowing what to do with it, if you're someone like me who isn't an expert in ODEs. Do you know of any accessible resources on spectral methods?

(Of course arbitrary precision is far, far worse than double precision in terms of performance, which is why so much work goes into techniques that avoid increasing precision. But sometimes you have few enough instances to solve that you can simply eat the cost.)