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It is somewhat similar to the percolation theory, but I can't find any reference for what happens with multiple colors involved.

Let there be an infinite grid and $n$ colors. Each tile in the grid is assigned a random color (each color with probability $1/n$). What is the average size of all unicolored areas? (We always consider the biggest possible unicolored area).

I am most interested in a solution for $n = 3$.

I am worried that there will not be any nice closed form solution, but I do hope that it will be something beautiful involving $e$.

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    I highly doubt that a closed form is possible. If you have programming skills, and if the problem intrigues you, you might consider writing a program to get an approximate answer via a simulation. – 2017-02-28
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    Actually, now that I think about it, it's quite conceivable that the expected size of a unicolored contiguous region is _infinite_ (for n < 4, I see it as intuitively clear). If so, that might not be so hard to prove. – 2017-02-28
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    @quasi I made a simple simulation with javascript, and got consistent results of average size of an area ~2.69 (with max size consistently over several runs around 55-60 on a 1000x1000 grid). Maybe I am misunderstanding the meaning of the words expected size, but I think you are wrong. For n = 2, I got ~7.61 average size. – 2017-03-01
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    Ok, well data disproves flawed intuition, so I'll defer to your results. – 2017-03-01

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It really is a case of (site) percolation. Let $A$ be the largest connected unicoloured region containing a particular tile. This is equivalent to the percolation cluster containing an open site, where each site is open with probability $1/n$. You just identify an open site with a tile of the same colour as your particular tile.

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    That does not really answer the question of what is going to be the average size of the regions, only the expected size of a region which includes a given point. These are two different things (for example, when you have a big unicolored region, there are more likely to be "holes", which create very small unicolored regions, which you do not account for with your solution) – 2017-03-01
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So I made some simulations and dug a bit into this. Let's see whether someone stumbles upon this question.

The answer for n=3 is ~2.695. It is very close to $e$, but not quite. The average size only rarely exceeds $e$ with simulation on 1000x1000 grid. It indeed is very consistently around 2.695. I don't have a closed form solution, and I am now convinced that there is not one.

Average area for n=2 is ~7.55 and for n=4 is ~1.94

Read more in my blogpost.