I'm a university student in my 2nd year doing a double major in political science and history and have been personally researching and reading/watching news, information and articles on world militaries also learning from my courses and from several websites I gathered (globalfirepower.com, globalsecurity.org, nationmaster.com, cia.gov, etc.) here are my top 10 and extended Top 15 militaries in the world according to my studies and research.
1.) The United States of America - Obviously and undisputed top military in the world due to the second standing largest army and very powerful combined force of army, navy and air force, excellent military training with their marines, delta forces, seals, excellent technology and economy, numerous allies, more than 100 military bases and global influence in the world, 2nd largest nuclear arsenal (however most technologically advanced nukes), huge military budget, extensive and continuing military experience in wars such as Iraq, Afghanistan, Pakistan and now in Yemen, permanent member of UN Security Council, 4th largest country in the world by area/land mass and 3rd by population with over 300 million people, one of the largest military weapons technologies and producers in the world, leading member of NATO, etc.
2) People's Republic of China - largest standing army in the world as well as the largest country in the world with a population of over 1.3 billion, equally large and powerful navy and air force, good military training, improving navy and air force, between 200-300 or more nukes making it between 3rd-5th largest nuclear arsenal in the world, second highest military budget after US, growing economy and rapidly improving technology, self-sufficient in military production and huge military weapons technology, permanent member of the UN Security Council, most powerful Asian military, etc.
3) The Russian Federation - 5th largest army in the world with over 1,000,000 troops, largest nuclear arsenal in the world (one of the two most advanced nukes in the world, the other being US), largest country by area/land mass by far, one of the largest countries as well with a population of over 142 million people, most powerful Eurasian country, very disciplined and professional military training, large and powerful navy and air force as well, one of the best military technologies and weapons producers in the world, stable and growing economy after the Soviet collapse in 1990, permanent member of the UN Security Council, large military budget, one of the most experienced militaries in the world, etc.
4) Republic of India - 3rd largest standing army in the world after China and US, second largest country in the world by population after China with over 1.75 billion people, also one of the largest countries in the world by area/land mass, one of the largest and rapidly improving and powerful army, navy and air forces in the world, good military training, experienced military with wars over Pakistan and China, growing military weapons production and technology/industry, large military budget, 3rd most powerful Asian military power, etc.
5) The United Kingdom - one of the largest military budgets and economies in the world, small combined military force of army, navy and air force personnel in the world but very effective nonetheless, the most powerful navy and air force with a strong army as well in Europe and one of the best and largest navies and air forces in the world as well, permanent member of the UN Security Council, one of the leading members of NATO, one of the largest countries in Europe with 62 million population after Russia with 142+ million, Germany 80+ million, Turkey 75 million and France 65+ million, one of the best military training in the world along with US, Canada, Australia, Russia, etc., an experienced military with recent NATO involvement in Iraq and Afghanistan, one of the top nuclear powers, one of the leading NATO and EU members, etc.
6) France - permanent member of the UN Security Council, one of the best militaries in Europe and very good military training, decent sized and very powerful army, navy and air force, one of the largest economies and military budgets in the world, one of the largest countries in Europe with over 65 million people, one of the top nuclear powers, one of the leading NATO and EU members, etc.
7) Germany - largest country in Europe with over 80+ million people, one of the leading NATO and EU members, excellent military training and one of the best military weapons technology and industry in Europe and the world, 4th largest economic power in the world and one of the best technologies in the world, decent sized but very powerful and professional combined force of army, navy and air force, one of the best tanks, air craft carriers and fighting planes, and other military weapons producers in the world, not a declared nuclear power but is a member of the NATO nuclear sharing policy and possesses a number of nuclear weapons along with Turkey, Italy, Belgium and Netherlands, one of the leading members of NATO and EU being the most populous member of EU, a defensive force but in war time can mobilize and possess a very large and powerful army and overall military, is very capable of producing its own nukes if need be but due to its bad reputation in WWI and especially WWII not allowed to acquire its own produced nukes, etc.
8) Japan - one of the best navies and air forces in the world as well as a small but very professional and well trained army, 2nd largest economy in the world and one of the most technologically advanced countries in the world, one of the most populous countries in Asia and the world with over 127 million people, a defensive force like Germany but can quickly create a very large and powerful army and overall military if need be, does not possess any nuclear weapons but very capable of producing its own, due to their bad reputation in WWII like Germany restricted military, etc.
9) Republic of Turkey - most powerful military of Middle East and Central as well as Western Asia, 2nd largest army and overall military of NATO after US, excellent military training and very well trained according to NATO standards, a very large (over 1,055,000 overall military) and powerful combined force of army (514,000 active, 380,000 reserve and 144,000 paramilitary), navy (best in Middle East and the Muslim world, and air force (2nd best in Middle East after Israel), one of the largest Eurasian countries with 75 million people, largest economy in Middle East and in the Muslim world being 17th economy in the world in GDP nominal and 15th in GDP PPP, 8th largest military budget in the world, technologically and in military weapons industry the best in Middle East and in the Muslim world as it has access to NATO weapons and produces 60% of it's own weapons and has future impressive military production projects like a native tank based on the German Leopard model jointly created by South Korea, a unique and strategically as well as geographically important country and considered a bridge between East (Asia and Russia) and West (Europe and the Americas) being near Middle East, one of the few Muslim majority (99% Muslim) countries but a secular democracy, has great influence in Middle East and in the Muslim world due to it's long legacy of Turkic empires ruling over most of Muslim lands especially from the previous Ottoman Empire (1299-1923 ruling over three continents of Arabia and Middle East, North Africa, Anatolia/Asia Minor of current day Turkey and Southeastern Europe/Balkans as well as Caucasus), part of the NATO nuclear sharing policy and possesses 90 nuclear weapons (40-45 of them under the personal use and disposal of the Turkish air force and navy), is capable of producing it's own nuclear weapons with a growing economy and military technology, has signed a deal with neighbouring Russia to open Turkey's first nuclear power plants in the next few years after neighbouring Iran's attempts to acquire these weapons to maintain a balance of power in the region, part of G20 major economies in the world, produces food, weapons, automotives, textiles and clothing, very attractive tourist location, etc.
10) Israel - one of the best military trainings in the world, a very experienced army with it's several wars with the neighboring Arab countries since it's formation in 1948, best air force in Middle East, very self-sufficient in creating its own military weapons production/industry/technology, small but very professional and powerful army and air force and decent navy, a nuclear power, strong alliances with US, NATO/EU/Europe and Turkey, many resources and self-sufficient in these vast resources, one of the most experienced militaries in Middle East, the Muslim world and in the world in general with the Turkish Cyprus invasion in 1974, and the recently continuing civil war in the eastern part of Turkey with the separatist/communist/terrorist organization called PKK (Partiye Karkireye Kurdistan meaning Kurdistan Workers Party) for 25+ years since 1984-present, etc.
11) Islamic Republic of Iran - very good military training and professionalism, one of the top 3 largest military powers in Middle East the Muslim world and 8th in the world, large and powerful army and air force, decent navy, seeking to develop and acquire nuclear weapons, the major Shiite Muslim country, over 70 million population being one of the largest Muslim countries, strong alliances with Russia and China, an experienced army with it's recent war with Iraq from 1979/80-1988/89/90, vast reserves of oil and gas, etc.
12) Islamic Republic of Pakistan - very good military training and professionalism, the largest Muslim military power in the world with over 600,000 active troops and the 6th/7th largest in the world, very large and powerful army and air force decent navy, a very experienced army with it's wars with neighbouring India, the only Muslim nuclear power, strong alliances with Turkey and the rest of Middle East and the Muslim world especially with the Arab states, China, France and some NATO/EU/Europe, and US, largest contributor to UN peacekeeping missions with over 10,000 soldiers, second largest Muslim country and one of the largest in the world in population with 180 million people, etc.
13) Brazil - a growing military and economic power in the world, largest and most powerful military and economy in South and Latin America, strong alliances, one of the largest countries in the world by population (over 192 million people) and area/land mass (5th/6th largest in the world), etc.
14) South Korea/Republic of Korea - one of the most economically and technologically advanced countries in Asia following Japan's model, 6th largest active army in the world, a very powerful and large army, navy and air force, one of the best navy and air forces in Asia, very self-sufficient in military weapons production/industry/technology and general production, over 45 million people, strong alliances with Japan, Turkey, US and NATO/EU/Europe, etc.
15) North Korea/Democratic People's Republic of Korea - the most militarized and combat ready army in the world, the 4th largest active army in the world, a very large and powerful army and air force decent navy, over 23 million population, etc.