Has anyone got a smart idea as to enlarging a hole in a foam sheet like this? 

Must be perfectly circular, perfectly central and within a few tenths of a millimetre.  Free hand is definitely out!  

I was hoping that cookie cutters could be bought with exact mm increments but I haven't been able to find a set so far.

Hole saw?

A google image search had some specifically for foam without the teeth. I don't know if those would work better.

I've also seen someone cut teeth on the end of a bit of PVC pipe:

Ah, here's the blog post about that: https://davidneat.wordpress.com/2016/01/17/cutting-smooth-holes-in-foam/

How you'd ensure it was sufficiently central though I'm not sure, you'd probably have to make a jig to guide it.

Maybe with a jig you could use a router or dremel or something, if finding something above with the correct dimensions doesn't work out?

Or invest in a fancy laser cutter 😈

Oh another idea - I'm not sure how practical this might be, but if you had a round bit of wood like a broom handle, and happened to have a lathe, you could cut it down to the perfect size, with a point on it, and glue fine sandpaper to it, and then as you spit spin it and lower it through the hole you'd gradually grind away the material to the correct diameter. Could work maybe??Not sure how practical though 🤷‍♂️

Oh, yet another idea - Hot wire? Could you rig up a thin straight bit of wire with some current running through it to make it hot, and rotate the wire or foam with a jig such that you cut a nice neat circle of exactly the right size?

Thanks Lyndon.  That's definitely the engineer's answer.  I'm looking more for the lazy persons answer!

I assumed all hole-saws had teeth that would rip the foam.  If some don't then I'll have to check prices for those in the morning.

For context, I could buy a pack of 10 in the correct size for £6.  But I don't need 10 and already have some smaller ones spare.  So was hoping for some sort of multi-size tool to create the sizes I need as and when necessary, rather than buying a pack of each size unnecessarily.

The example you illustrated was for coin holders, I'm sure you know you can buy a wide variance of these...

Just now, StephenFord said:

The example you illustrated was for coin holders, I'm sure you know you can buy a wide variance of these...

Yes.  But those are made of cardboard and cling film.  They look awful.  (To me at least!)

Appreciate you posted at the same time but I know I can buy the foam version in packs of 10 for £6.  I just don't need 10 of each and thought this would be a more efficient way to go.

Looks like proper foam hole saws are super expensive, and more suited to a commercial need.

This is the type of thing I'm after, but these ones aren't small enough or in millimetre increments.  Any other suggestions for something like this but smaller before I give up and just buy the correct sized foam lol?

12Pcs Hole Hollow Punch Cutter Set With Storage Case Stainless Steel Round Cutting Mold Manual Hole Puncher Set - AliExpress 15