<i>Doubleplus</i> in <a href="https://japanese.stackexchange.com/a/32178/21641">this context</a> means "extremely", probably with an implicit suggestion that the organization being spoken of is oppressive or is part of an oppressive and unjust system.

The word <i>Doubleplus</i> is not standard English.  It was coined by George Orwell in his book 1984.  In that book, the government manipulates the language, making it simpler and simpler with the goal of making it difficult for its populace to express ideas that the government does not approve of.  Because the word is not standard English, you may want to avoid using it yourself.  Even some native English speakers will find the word confusing or strange.

The position of the quotation marks makes a difference, too.  The quotes in the original comment were around the phrase "doubleplus military", meaning the phrase "doubleplus military" made the commenter smile (rather than the military itself making the person smile).