In the DS9 episode "Whispers" - O'Brien orders coffee "double strong, double sweet". The double sweet is relatively obvious. In-universe, what does 'double strong' mean? Is this coffee brewed/replicated with an extra dose of caffeine, or extra tannins, 50% less water, or double shots of espresso, other alien stimulants, etc...?
"Double strong" means it's been brewed twice (emphasis mine)
Like a strong-tasting cup of joe, but want more caffeine out of it? Start double brewing your coffee.
Though it sounds both simple and complex, double brewing your morning coffee doesn't require much beyond its name: just send your coffee through another brew cycle for a double-strong cup of joe.
It means a stronger taste and more caffeine, generally. There's nothing to suggest that O'Brien was ordering something different than this.
From a practical standpoint, this is more likely to be (replicated) "red-eye" than actually brewed a second time (by drip, press, or other method).
What's a "red-eye"? According to my local barista, it's a cup of coffee with a shot of espresso added. No reheating to damage the original coffee, but approximately double the caffeine, and roughly double the other dissolved/suspended "coffee stuff".
This seems to be a very strong reference (or pun) to the novel 1984, where the antagonist is also called O'Brien, a high-ranking member of the ruling totalitarian party, which has among its goals a simplification of the English language, in order to make expressing emotions more difficult. Among these reforms is the changing of the adjectives "good, better, the best" into things like "good, plusgood, doubleplusgood". So "double-something" in that context is nothing more than expressing a superlative adjective (the strongest coffee, or, at least, an extremely strong coffee).