Mr. James Grime from Numberphile channel has said () that the Euler´s number e has basically nothing to do with geometry.
I would like to know if there is any derivation of e based on geometrical arguments.

Probably not what you're looking for, as it is no construction method (as there can't be any with compass and ruler). But you can define ##e## to be the right boundary of the area under the standard hyperbola with its left boundary ##1## that equals ##1##.
$$ 1 = \int_1^e \frac{1}{x}dx$$

This kind of geometrical association seems to be more inclined to the calculus itself.
Like e is the value of a base ## b ## such that ##b^{i \theta}## corresponds to the complete unit circle as ## \theta ## goes from ## 0 ## to ## 2\pi ##.

How would you answer your question if it was ##\pi##? This is clearly a geometric number, but cannot be constructed either. It's the relation between diameter and circumference of a circle or the area of a circle to its squared radius. Circles and hyperbolas are closely related, so I find the definition above quite similar.

Radius, length, areas (total area), curvature are kind of anatomic parts of geometrical objects. Taking a function and doing a definite integral from a to b is a method which can generate almost anynumber, don´t you agree?

No, I don't. The integral was simply shorter to write than to make a picture of the area. You can also generate any number as the area of a circle which is also an integral.

In the case of circle one can speak of total area. Regarding the hyperbola, it doesn´t happen. I see some difference, but I agree that there is some geometry involved here.

I admit, ##e## is kind of hidden in comparison to ##\pi## and I wouldn't call it a geometric definition as you required. But at least it's not completely out of business.

The first definition they give, for a logarithmic curve, is ##r = ae^{b\theta}##. The next equation they give is for ##\theta##, in terms of the natural log function, ##\ln##. The parametric equations are given in terms of exponential functions involving e.

Dear Mark, we are speaking of e appearing as a singular real number in a geometrical context. Statements where e can be exchanged by any other positive real number does not seem to help.
We may say that it was only a matter of the author´s preference.

I am satisfied with the contributions. It is perhaps the case that the question per se is not so well formulated. I am sorry. You may end the thread. I also believe the answer to the OP is yes.