Non-interesting numbers via resource-bounded properties?
This might be cheating, because I am reinterpreting your notion of "number" as "countable ordinal."  However, there is an extensive theory about exactly what you are asking about in your question -- which ordinals can be defined, with which definition-strengths.  I looked at Wikipedia, and the large countable ordinal page has a lot of information.  In particular, the "smallest number that is interesting because the system can't even prove it's a number" corresponds to the unrecursable ordinals, if we require nothing other than computability as the resource bound.