That's a broad question, but I can throw some thoughts at you from personal experience.
I'm actually an Australian who has worked in a couple of companies but across multiple countries and I've found out first hand that you have a wealth of opportunities that other people don't have, but you also have a lot of problems that other people won't have.
First up, asset classes. Real estate is a popular asset class, but unless you plan on being in each of these countries for a minimum of one to two years, it would be seriously risky to invest in rental residential or commercial real estate. This is because it takes a long time to figure out each country's particular set of laws around real estate, plus it will take a long time to get credit from the local bank institutions and to understand the local markets well enough to select a good location.
This leaves you with the classics of stocks and bonds. You can buy stocks and bonds in any country typically. So you could have some stocks in a German company, a bond fund in France and maybe a mutual fund in Japan. This makes for interesting diversification, so if one country tanks, you can potentially be hedged in another.
You also get to both benefit and be punished by foreign exchange movements. You might have made a killing on that stock you bought in Tokyo, but it turns out the Yen just fell by 15%. Doh.
And to top this off, you are almost certainly going to end up filling out tax returns in each country you have made money in. This can get horribly complicated, very quickly. As a person who has been dealing with the US tax system, I can tell you that this is painful and the US in particular tries to get a cut of your worldwide income. That said, keep in mind each country has different tax rates, so you could potentially benefit from that as well.
My advice? Choose one country you suspect you'll spend most of your life in and keep most of your assets there. Make a few purchases in other places, but minimize it. Ultimately most ex-pats move back to their country of origin as friends, family and shared culture bring them home.