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Is there a comprehensive list of real-number-describers that allude to the properties of that number?
For example:
Amicable numbers, Abundant, Deficient, Perfect, Carmichael, prime, transcendental, etc.

1. If so, where can I find it?
2. What are some new, exciting types of numbers with interesting properties?
3. What is name of the type of irrational number that contains every natural number as a substring of its decimal expansion ?

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for three: normal – user58512 Jan 24 '13 at 18:26
for two: liouville – user58512 Jan 24 '13 at 18:26
@user58512, all normal numbers satisfy 3, but not all numbers that satisfy 3 have to be normal. – Karolis Juodelė Jan 24 '13 at 18:27
I wouldn't say the Liouville numbers are a new type, as asked in 2. After all, Liouville has been dead for over a century. – Andreas Blass Jan 24 '13 at 18:29
for three: "disjunctive number" and "rich number in base 10" seem to be somewhat in use at present. – Dave L. Renfro Jan 24 '13 at 18:42
up vote 0 down vote accepted

Wikipedia provides a solid list of numbers:

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This is a good list, @Jeremy Thank you. I do feel like there are more than this list has though. – Rustyn Jan 24 '13 at 18:57
@Rustyn Yazdanpour: Some that might not be on that list (I haven't checked) can be found by googling "transcendence measure" number, "computable real number", "pythagorean constructible"‌​, "explicit algebraic number", and "closed form number". – Dave L. Renfro Jan 24 '13 at 20:01

For 3, the general type is a normal number, though this is more restrictive than your request. These will have all sequences present in the proper proportion. The specific one $0.123456789101112131415161718 \ldots$ is known as Champernowne's constant

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Hey thanks for your contribution, @Ross – Rustyn Jan 24 '13 at 19:00

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