Which general methods are there for determining if $t$ and $e^t$, where $t$ is any given transcendental number, are algebraically independent?

Question

Which general methods are there for determining if $t$ and $e^t$, where $t$ is any given transcendental number, are algebraically independent?

Can Schanuel's conjecture be used for this?

We have Lindemann-Weierstrass theorem, Gelfond-Schneider theorem and Schanuel's conjecture.

But can these used for the two transcendental numbers without using algebraic numbers?

For Lindemann-Weierstrass theorem and Gelfond-Schneider theorem, algebraic numbers are a prerequisite of the theorem.

I want to determine the following transcendence degree: $$\text{trdeg}_\mathbb{Q}\mathbb{Q}\left(t,e^t\right).$$

But Schanuel's conjecture states only the lower limit $1$ of this transcendence degree.

My experience is:
If we want to use Schanuel's conjecture for determining the exact value of the transcendence degree, we have to give

• at least one non-zero algebraic number

  or

• a tower of elements whose tower of exponentials contains at least one non-zero algebraic number.

Are there further strategies for using Schanuel's conjecture?

Examples:

Let $a$ be a non-zero algebraic number.

The semicolon indicates that the exponential pictures follow.

$$\text{trdeg}_\mathbb{Q}\mathbb{Q}\left(a;e^a\right)=\text{trdeg}_\mathbb{Q}\mathbb{Q}\left(e^a\right)=1$$

$$\text{trdeg}_\mathbb{Q}\mathbb{Q}\left(\ln(a),a;a,e^a\right)=\text{trdeg}_\mathbb{Q}\mathbb{Q}\left(\ln(a),e^a\right)=2$$

But how can we determine the exact value of the transcendence degree $$\text{trdeg}_\mathbb{Q}\mathbb{Q}\left(t,e^t\right)?$$

I need that to find an answer for How to prove or disprove algebraic independence of exponential numbers?