Completing a game is far by the most difficult thing to come across in a standard Pokémon came. This is due to the presence of many reasonable objectives to finish. For example, I take the very first game I have ever devoted reasonable progress in the hope of completing it, Pokémon Ruby. I can list out the objectives as:
★Beating the Elite Four - ★
★Getting Master Ribbons in all contests - ★★★★
★Getting through a 100 win-streak in the Battle Tower - ★★★
★Completing the Regional Pokédex - ★★★★
★Completing the National Pokédex - ★★★★★
As you can see, not all of the objectives are simple, some are fairly hard, some even downright impossible. Which is why it is extremely rare to come across a completed game. Thus, I classify my qualifications of 'completing' a game as:
★Beating the Elite Four (twice, if needed)
★Completing the Post-Game Storyline
... and that pretty much sums it up.