It is possible that in the present day it is. I am sure that such a statement would be contentious at the very least.
However, historically the US military was very much not "the greatest fighting force in the history of mankind."
Historically, that is until the post Vietnam era, the United States has had a strong aversion to a professional military. It has been the habit of the United States to only build a large army and navy during a time of war, and to disband most of those forces after each war.
About a year ago myself and several other users discussed the habit of the US military to go through this up and down pattern of troop and equipment strength. Make sure to read the child comments by /u/Cenodoxus and /u/Irishfafnir as well, good stuff there.
Maybe. This question is perhaps not best in this subreddit as it's really more about the current state of various armies than about history, but I'll do my best to answer it.
Right now, the United States Army is easily the most powerful military force on the planet. They can deploy troops in force anywhere in the world and can muster way more firepower than any other force in the world as well as being significantly more technologically advanced than all of its rivals. Just as an example, in the initial stages of the Iraq War (against Saddam) America suffered fewer than 100 casualties).
So how does this stack up historically? I'll try to focus on the areas I know the most about, but history is very long and complicated :p and I'd welcome counter-examples and the like.
The civilizations that come to mind when I think of military forces that are significantly more powerful than all other military powers near them in the sense that they can deploy troops anywhere and "outgun" their opponents are Rome and The Mongols under Ghengis Khan. Let's compare the militaries of these nations to the present United States Army:
Rome: Roman tactics and military engineering were far ahead of their time. Roman troops drilled constantly, which gave them a massive advantage over other armies as individual Roman units could precisely execute complicated maneuvers even in the middle of battle (ie. The tactic of rotating troops through the front line of a unit every 30-45 seconds which was used to great effect at the Battle of Watling Street). Roman Legions from Italy could also rely on minor military innovations such as the Pilum which proved to be consistently effective. That being said, Roman Legions were hardly homogeneous forces (as talked about in this askhistorians thread) and tactics, training levels, and tech differed between legions and formations, which points towards the Roman advantage coming from something other than doctrine and tactics as these could wildly vary.
Mainly, Rome’s advantage came from their highly advanced logistics system. Roman roads allowed rapid redeployment of military forces and easy shipment of supplies and the Roman Army had institutionalized systems for distributing supplies and other material among the troops that were incredibly advanced compared to contemporary systems (at least in the early empire). Compared to the Greek City states which expected soldiers to pay for their own supplies, Rome’s grain rations gave them a massive advantage as it meant the legion could operate longer on steady supplies. See this book for more details on Roman supply. I’d say Rome’s ability to consistently dominate other nations was a result of their advanced logistics system which allowed them to redeploy troops (relatively) easily and with great speed around their empire as well as keeping them in supply for long campaigns. Notably however, Rome was hardly completely dominate. The Persians for instance, were able to consistently defeat Rome and proved pretty much immune to invasion as Roman supply lines were too stretched to allow a march into Persia.
Logistics is one of the major strengths of the American Army as well. It’s difficult to describe how much America DOMINATES the logistics systems of other nations. Neither China nor Russia can deploy large amounts ground forces farther away than a neighboring country while America can maintain a couple hundred thousand troops on another continent for more than 10 years (one hundred thousand of those troops in an inland country). America’s ability to deploy troops around the globe far surpasses that of their rivals. America’s military has a bigger advantage in logistics than Rome did. I would also argue that on a tactical level, America’s army has a bigger advantage in pure firepower than Rome did.
Mongolia: Steppe tribe’s armies being superior to their “civilized” counterparts was not new too Mongolia’s rise. Steppe tribes had consistently dominated their opposition (see The Huns) and were mostly not a threat simply because they were too divided or not interested in conquest. What made a Steppe Tribe army superior? Fast movement (entirely mounted force), elite troops (soldiers had been training as horsemen and bowman their entire lives), and very little need for supplies (Steppe could live off the land and the soldiers themselves had been living off the land their whole lives).
To this winning formula, Genghis Khan introduced an excellent base ten system of organization and a unified command structure. Mongolian troops simply dominated their opponents. That’s how Genghis Khan and his successors built the largest land empire in history. On a tactical level, troops were more disciplined and could throw out more firepower while on a tactical level Mongol armies moved independently of supply lines.
If I were forced to pick an army with an advantage over its opponents similar to the ones currently enjoyed by the US army I would pick Mongolia. It had the firepower advantage and the logistics advantage currently enjoyed by the US army. However, I don’t think that Mongolia is quite there yet if we include the full US military. Biggest example is that Mongolia lacked a significant Navy, which is an area that America completely and utterly dominates its opponents in.
So, Yes, I think there is truth to what the US Army says. Notably however, this is changing as the modern world changes to a more multi-polar one and countries like China and Russia build up.
This submission has been removed because it violates the rule on poll-type questions. These poll-type questions do not lend themselves to answers with a firm foundation in sources and research, and the resulting threads usually turn into monsters with enormous speculation and little focussed discussion. “Most”, “least”, "best" and "worst" questions usually lead to vague, subjective, and speculative answers. If you'd like, you may PM /u/caffarelli to have your question considered for an upcoming Tuesday Trivia thread.