Supertranscendental numbers?

Factual Questions
1

Transcendental numbers are those which can not be produced by a finite number of arithmetic operations. (I mean by this addition, subtraction, division, multiplication, exponentiation, and the taking of roots. I don’t know if “arithmetic operations” is the correct label for this.)

Are there any real numbers which can’t be produced even by an infinite number of arithmetic operations?

Is there a name for such numbers, if they exist?

Does it make any difference if I take out the stipulation that they be real numbers?

-FrL-

2

I make mistake in above post.

Transcendental numbers are the ones that are not the solution to any polynomial with integer coefficients.

I don’t know what I was thinking of.

Sorry, never mind. (Unless you have some idea what I was thinking of.)

-FrL-

3

No, because they could be written as the sum of an infinite series. Pi, for example, = 3 + 1/10 + 4/100 + 1/1000 + 5/10000 + etc.

4

No, all real numbers can be produced with an infinite number of operations (here I’m taking infinite to mean “countable”).

In particular, the number 0.abcd… (where each letter represents a digit) is the countable sum .a + .0b + .00c + …

If we still take “infinite” to mean “countable”, I’m actually not entirely sure. I would expect there are probably cardinals that satisfy what you’re looking for. On the other hand, this might be getting in the realm of “large cardinal” axioms, and it might be the case that the existence of such cardinals is independent of standard (ZFC) set theory. (I’m not sure off the top of my head, and I don’t have my set theory book handy at the moment.)

5

I just realized it might not be clear, but my last paragraph was responding to “Does it make any difference if I take out the stipulation that they be real numbers?”