What motivates the arithmetic-geometric mean? What motivates the arithmetic-geometric mean? What inspires it? I understand how to calculate this mean but do not understand what might prompt a mathematician to pursue such a mean in the first place.
Is there a notion or mental image that conjures this thing?
(See also this related question. If it helps you to pitch your answer at the right level, I am a STEM professional but not a professional mathematician.)
 A: The Wikipedia article Elliptic integral states

The complete elliptic integral of the first kind is sometimes called the quarter period. It can be computed very efficiently in terms of the arithmetic–geometric mean:
  $$ K(k)=\frac{\frac\pi2}{\text{agm}(1,\sqrt{1-k^2})}. $$

Thus, the motivation is to compute complete elliptic integrals. The Wikipedia article Arithmetic-geometric mean states

The first algorithm based on this sequence pair appeared in the works of Lagrange. Its properties were further analyzed by Gauss.

It was used extensively by Adrien-Marie Legendre in
computing his tables of elliptic integrals contained
in Traité des Fonctions Elliptiques, published in 
three volumes 1825, 1826, and 1830.
So much for historical facts. Your question was

Is there a notion or mental image that conjures this thing?

and this is very rarely recorded. The facts indicate that
there was a need to compute complete elliptic integrals,
and somehow one or more mathematicians were able to develop
an equation which related elliptic integrals with different
parameters. The first instance seems to be due to John Landen.
The DLMF section 19.8(ii) Landen transformations has the following equations

Let $$ k_1 = \frac{1-k'}{1+k'} $$
$$K(k) = (1+k_1)K(k_1), \qquad E(k) = (1+k')E(k_1)-k'K(k).$$

You can refer to the Wikipedia article Landen's transformation for
more details.
