Let's see... I was born in 1980, and USSR and Ukraine employed (in case of the latter, still employ) draft, so there would have been a chance of me ending up in the military if I had not moved to the US in 1996, albeit fairly small unless there was a war where they would draft college students (which they do not do).
Now, alternatively, say I not only end up staying in the Ukraine, but end up enrolled in the military academy instead of civilian university. Thus, by the time I am in my early 20s, I am an officer in Ukrainian military. Come 2003, and US-led invasion of Iraq proceeds as in OTL, with small number of Ukrainian troops (including yours truly) sent in afterwards to hold some apparently strategically unimportant place as a part of the "coalition".
While there, I end up a commander of small regiment on patrol that is ambushed by a superior number of insurgents, and get to be lucky enough to not only survive the shootout, but to also destroy or capture all of enemy combatants while suffering little or no losses. Repeat it a few times, and by the time I return home, I am a decorated war hero...
Of course, there is a usual scenario of joining the army/navy/marines/air force, and ending up in Iraq or Afghanistan - bonus points for taking part in capture of one of bin Laden's top cronies or even the man himself, but that is way too close to OTL... I would have been 22 at the start of Iraqi war, so it would have been just the right age to end up in the Gulf had I ended up in the US military for whatever reason (although I would have to have been really disillusioned at the time with college, my career, my music, relationships, and just about everything else to decide to "screw it all" and go off and join the military).
Or, another scenario. The relatively peaceful initial breakup of USSR leads to war between Ukraine and Russia some time between 1994 and 1996, possibly over Crimea or Black Sea Fleet (which has been feared in Ukraine at the time might come to pass). Things go so badly for Ukraine that they have to draft 14-16 year olds as Russia simply has much greater numbers, and, in TTL they are hell-bent on retaking Ukraine through force. Give it a year or two, and I can distinguish myself somehow... there it is, a fresh-made military hero that is not even legally able to buy cigarettes, let along alcohol.