Closed-form numbers

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There are many variations of the concept of closed-form numbers.

Elementary numbers

Here, "closed-form" implies "explicit" or "implicit."

(Chow, 1999):

(...) Ritt thought of elementary numbers as the smallest algebraically closed subfield $\scriptstyle \mathbb {L} \,$ of $\scriptstyle \mathbb {C} \,$ that is closed under $\exp$ and $\log$. (...) "elementary numbers" are now numbers that can be specified implicitly as well as explicitly by exponential, logarithmic, and algebraic operations, and $\scriptstyle \mathbb {L} \,$ is now sometimes called the field of Liouvillian numbers.

The algebraic numbers form a subfield of the elementary numbers.

EL numbers

Here, "closed-form" implies "explicit."

(Chow, 1999):

Definition. A subfield $F$ of $\scriptstyle \mathbb {C} \,$ is closed under $\exp$ and $\log$ if (1) $\scriptstyle \exp(x)\,\in \,F\,$ for all $\scriptstyle x\,\in \,F\,$ and (2) $\scriptstyle \log(x)\,\in \,F\,$ for all nonzero $\scriptstyle x\,\in \,F\,$, where $\log$ is the branch of the natural logarithm function such that $\scriptstyle -\pi \,<\,\Im (\log x)\,\leq \,\pi \,$ for all $x$. The field $\scriptstyle \mathbb {E} \,$ of EL numbers is the intersection of all subfields of $\scriptstyle \mathbb {C} \,$ that are closed under $\exp$ and $\log$.

(...) The "EL" in the term "EL number" is intended to be an abbreviation for "Exponential-Logarithmic" as well as a diminutive of "Elementary"; it reminds us that $\scriptstyle \mathbb {E} \,$ is a subfield of the elementary numbers. (...) $\scriptstyle \mathbb {E} \,$ is countable (...) $\scriptstyle \mathbb {E} \,$ admits an explicit finite expression in terms of rational numbers, field operations, $\exp$ and $\log$.

The arithmetic numbers ("explicit" closed-form algebraic numbers) form a subfield of the EL numbers. (The nonarithmetic algebraic numbers are excluded.)

Closed-form numbers not involving $\exp$ and $\log$

What about "explicit" or "implicit" closed-form numbers not involving $\exp$ and $\log$, i.e. numbers that admit an "explicit" or "implicit" finite expression in terms of rational numbers and field operations?

The algebraic numbers form a subfield of the closed-form numbers not involving $\exp$ and $\log$.

Explicit closed-form numbers not involving $\exp$ and $\log$

What about "explicit" closed-form numbers not involving $\exp$ and $\log$, i.e. numbers that admit an "explicit" finite expression in terms of rational numbers and field operations? For example,

$\scriptstyle 2^{2^{1/2}}\,$

is a transcendental number.

The arithmetic numbers form a subfield of the "explicit" closed-form numbers not involving $\exp$ and $\log$. (The nonarithmetic algebraic numbers are excluded.)