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It's a little tough to give examples of nondefinable numbers in a finite amount of space

You, sir, have a knack for understatement.
On the other hand, you're referring there only to mathematical definitions, so there might be some examples. It's conceivable, for instance, that alpha (the fine structure constant) is undefinable (in the mathematical sense), or any other dimensionless constant in physics (any of the mass ratios of particles, or chargetomass ratios, etc.). But I don't think that we could ever prove it to be so.
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