TLDR: YES
Yes, Amazon Echo will work outside of the supported countries, but some functionality will be lost.
Power
Amazon Echo supports 110-220 volts which means it works across the whole world without a voltage converter.
Plug
The Plug is localised to where ever you're buying it.

I had to buy an adapter.
App
Amazon Echo requires a companion App. But this isn't available in all areas. You'll have to jailbreak your iPhone or use an Android to install the app from the web.
The Android version is available: APK Mirror has a version
Services
Location based services won't work. The app expects an American 5-digit postcode. That means anything that requires your location might* not work.
This applies to core features:
- Local news
- Movie times
- Traffic
- Weather
This applies to skills as well, e.g.
- Uber
- Lyft
- TV guide
- Dominos
* See Below
What does work?
Core - Amazon itself works as long as it has power and internet.
Smart home - All the smart home devices that would otherwise work in your country work.
Time zones - You can set your timezone to anywhere in the world.
Music - It can play music if you would otherwise be able to play it, e.g. Amazon Play, Pandora & Spotify all work for me.
Weather - It can get weather for most cities in the world, but you must ask about your city. e.g. "Alexa, what is the weather in Cairo, Egypt?"
More Skills - All the skills that don't require your location have worked for me.