How to explain infinity to a $3^{rd}$ grader? In my country in $3^{rd}$ grade in math kids learn the four basic arithmetic operation (addition, subtraction, multiplication and divison) up to $10 000$.
My sister this year goes to $3^{rd}$ grade and one day she was writing her homework in math and out of a sudden she asked me up to which number a know to add, subtract... I answered that I know to add, subtract... for every number and because there are infinite amount of numbers, I know to how to calculate up to infinity.
This concept of infinity was unclear to her. He couldn't go over the fact that there are infinite amount of integers, because she thinks that ultimately there must be a largest number, one that's bigger of all of them.
I told her that because there are infinity amount of numbers I can always say a greater number than one she can told. She start saying $600000, 1245000000, 99999999999$ and I easily just added $1$ and obviosuly that makes my number bigger, but still it didn't helped her. I thought that just adding $1$ to hers humber will make her feel that she's close to beating me, so I though to double the number she says, but again it came with no success, because she stubornly continued "fighting with windmills".
How can I exlpain the existance of infinity to a 10 years old kid?
 A: A simple way that may help is explaining that infinity is just a word to describe that the number line never ends. Numbers will continue to get higher and just when you think you know the highest number, there is one number higher than that one. At this point, a ten year old does not need a specific proof or theoretical definition, but most importantly need something to compare it to, so they can visualize it in their head. 
A: You can try to ask her to provide a disproof rather than doing it yourself.  That is, if she comes and says "I've found the number $n$ which is larger than any number you've told me so far", then you could ask her "Do you think that is the biggest number possible or do you think I can come up with a bigger one?"  If she says that she thinks it is the biggest number possible, press her, "Are you so sure that you will perform chore $x$ if I can think up a bigger number?"
If she is the one who has the responsibility of always producing a bigger number, she may grasp the infinitude of whole numbers more easily.
I also agree with others that if she just does not get it at this point, it is no big deal.  It is a very subtle concept that the human brain is not particularly well equipped to grasp, something we forget after many years of mathematical training.
A: Say that
"infinity is like higher."
You can always go higher,
and you can always count one more.
Of course the next step
is the difference between
potential infinity
(you can always count one more)
and actual infinity
(the set of all integers).
There have been a few arguments
about this.
Then
you can talk about
different types of infinity.
The traditional way
is to ask if there
are more fractions than integers
and if there
are more points on a line
than there are integers.
Then comes Lebesgue measure,
but the child should
at least be in fifth grade for this.
At this point,
the child is ready for
graduate school.
A: Bishop Berkeley then later Poincare, Kronecker, Weyl, and Wittgenstein had trouble understanding the infinite, so give the 10 year old a break.
A: I know you've accepted an answer but because this popped up on the home page... You could always try explain with something physical, like something relate-able (ish). I have two approaches that come to mind at the moment.
The approach with numbers of "there is always a bigger number" if I pick $n$, $n+1$ is bigger again, which was frequently mentioned is akin to the idea of having a path/road that stretches far into the distance, to the horizon say. You can't see the end of it but for physical reasons you expect it should stop or end somewhere. You just do. People post about "infinity" being non-physical or real and maybe that's just in-built initially into how we think.
Which brings us to approach two: What if we have a path that loops, in a circle say or a figure 8? If we keep walking in one direction and keep following the path (there could be arrows on the path?), will we run out of path? What if we mark a start point and count how many times we pass it? So we relate it back to numbers this way.
Just a thought, kind of like the whole 8 on it's side thing $\infty$.
