My approach would be to ask them what it would mean to solve various equations. I am not sure what age we are talking about or what background knowledge they have.

The equations could include algebraic equations but more challenging would be to ask them what a number like $5^{\sqrt 2}$ is. They can probably type this into their calculators. You could ask them to solve $\sin(x)= 1-x^2$ (off the top of my head) or possibly $\exp(\exp(x))-\exp(x)+1=0$. The idea is to produce equations which obviously have solutions because you can plot graphs and see lines crossing but where it is also obvious that there is not going to be a formula for the solution.

This raises the problem of reconciling these two points of view. Then you resolve this by showing that you can find approximations to the solution and the approximations can be made better and better.

You could also discuss Archimedes approach to calculating the area of a circle.

The point is that you can work out the decimal expansion of "the number" to as many decimal places as you want.

Thanks to Pete for the link on what the question means. I was misled by the other responses.

Here are two thoughts:

i. What is the square root of 14? Well, it's between 3 and 4. Then we can do better by taking the average of 3 and 14/3. Then we can repeat this and do even better.

ii. Take the Fibonnaci sequence 1,1,2,3,5,8,13,... Take ratios of successive terms 1/1,1/2,2/3,3/5,5/8,8/13,... then these are approximations to the "golden ratio".

A personal anecdote: My daughter is 11 yrs old, bright and interested in maths. I tried i. on her and drew a blank look. I conclude from this that either it is a mistake to teach your own children or that she was not ready for this.