In my country, power comes in 220V by default. My original 3DS charger only accepts 110V power. I'm taking my 3DS everywhere, so I can't bring the big power converter I already have at home.
Is there any recommended small converter for converting 220V power to 110V?
Don't bother with a power converter unless you have a lot of appliances that need 220V→110V conversion. The Nintendo charger is itself a converter from 220V or 110V depending on the model, to the 4.6V used by the console. Chaining converters wastes energy, in addition to requiring two devices when one will do.
In my country, you can get a 220V 3DS charger for about $5 (example). Or, for about the same price, you can get a USB cable that lets you charge the console from any USB power supply, so you can combine that cable with any USB charger or PC that you have lying around.
As an alternative to a separate voltage converter (which for such a small device is probably overkill), although the 3DS states 4.6V as an input voltage, it will accept 5V.
Hence, you can simply buy a USB charger cable (like this one) and then you can use your laptop, desktop, or any of the myriad USB wall/car chargers that you can pick up cheaply (or like many, already have from your phone/tablet) that will work with 110/220V.
As long as you aren't trying to run a hair dryer off of it, this adapter should do the trick. It isn't grounded, which means it I would not expect it to do very intensive work, and some plugs may require you to fill the grounding peg hole to get a current. You should have some sense of supervision if you plan on drawing any more power from it. Though this adapter does have its problems, but it is quite lightweight and will get the job done for your purposes (and your 3ds).
Yet another answer is to use a 5V USB charger to charge your 3ds, and that will do a fairly good job without the need for a converter.
You can also just get a 220V 3ds charger/adapter, as they are very cheap.
