That's quite a selection of coins: Native American Nations. Hawaii. Libya (back when it was Tripoli). Samoa. The Philippines. China (during the Boxer Rebellion). Panama. Russia (during the Communist Counter-Revolution). Germany, Korea. Vietnam. Grenada. Iraq. Somalia. Bosnia/Herzegovina. Kosovo. Afghanistan. And that doesn't include the hundreds of little incursions around the world.

As for the nation we've invaded the most I'd say Mexico. Three times. The Mexican-American War. The Civil War occupation of Texas and blockade of the Nueces and Rio Grande Rivers. And the US occupation of Veracruz during the Mexican Revolution.
We've moved that border more times