Where does the battle of Armageddon take place?
The Battle of Armageddon takes place at a location called, in the Hebrew tongue, Armageddon, which is also described as the mountains of Israel and the valley of decision (the valley of Jehoshaphat).
Scripture names the site of this final great battle directly: the armies of the earth are gathered together into a place called in the Hebrew tongue Armageddon, as the sixth vial dries up the river Euphrates to prepare the way for the kings of the east to march there. This is where the beast, the false prophet, and the kings of the earth with their armies assemble to make war against Christ, who rides forth on a white horse as Faithful and True.
Ezekiel's prophecy of this same conflict locates it upon the mountains of Israel, where the great sacrifice of the slain armies is given to the birds of the air and the beasts of the field. Joel likewise speaks of this gathering, calling it the valley of Jehoshaphat and the valley of decision, the place where the Lord sits to judge all the heathen round about and where multitudes upon multitudes are drawn together for the day of the Lord.
After this war, Jesus sets His feet upon the Mount of Olives, the very place from which He ascended, and the mountain splits in two, forming a great valley. So the battle itself centers upon the mountains and valley region of Israel, gathered at the place called Armageddon, and concludes with the Lord's arrival at the Mount of Olives near Jerusalem.
The scriptures
And the sixth angel poured out his vial upon the great river Euphrates; and the water thereof was dried up, that the way of the kings of the east might be prepared. And I saw three unclean spirits like frogs come out of the mouth of the dragon, and out of the mouth of the beast, and out of the mouth of the false prophet. For they are the spirits of devils, working miracles, which go forth unto the kings of the earth and of the whole world, to gather them to the battle of that great day of God Almighty. Behold, I come as a thief. Blessed is he that watcheth, and keepeth his garments, lest he walk naked, and they see his shame. And he gathered them together into a place called in the Hebrew tongue Armageddon.
And I saw heaven opened, and behold a white horse; and he that sat upon him was called Faithful and True, and in righteousness he doth judge and make war. His eyes were as a flame of fire, and on his head were many crowns; and he had a name written, that no man knew, but he himself. And he was clothed with a vesture dipped in blood: and his name is called The Word of God. And the armies which were in heaven followed him upon white horses, clothed in fine linen, white and clean. And out of his mouth goeth a sharp sword, that with it he should smite the nations: and he shall rule them with a rod of iron: and he treadeth the winepress of the fierceness and wrath of Almighty God. And he hath on his vesture and on his thigh a name written, KING OF KINGS, AND LORD OF LORDS. And I saw an angel standing in the sun; and he cried with a loud voice, saying to all the fowls that fly in the midst of heaven, Come and gather yourselves together unto the supper of the great God; That ye may eat the flesh of kings, and the flesh of captains, and the flesh of mighty men, and the flesh of horses, and of them that sit on them, and the flesh of all men, both free and bond, both small and great. And I saw the beast, and the kings of the earth, and their armies, gathered together to make war against him that sat on the horse, and against his army. And the beast was taken, and with him the false prophet that wrought miracles before him, with which he deceived them that had received the mark of the beast, and them that worshipped his image. These both were cast alive into a lake of fire burning with brimstone. And the remnant were slain with the sword of him that sat upon the horse, which sword proceeded out of his mouth: and all the fowls were filled with their flesh.
Thou shalt fall upon the mountains of Israel, thou, and all thy bands, and the people that is with thee: I will give thee unto the ravenous birds of every sort, and to the beasts of the field to be devoured.
Proclaim ye this among the Gentiles; Prepare war, wake up the mighty men, let all the men of war draw near; let them come up: Beat your plowshares into swords, and your pruninghooks into spears: let the weak say, I am strong. Assemble yourselves, and come, all ye heathen, and gather yourselves together round about: thither cause thy mighty ones to come down, O LORD. Let the heathen be wakened, and come up to the valley of Jehoshaphat: for there will I sit to judge all the heathen round about. Put ye in the sickle, for the harvest is ripe: come, get you down; for the press is full, the fats overflow; for their wickedness is great. Multitudes, multitudes in the valley of decision: for the day of the LORD is near in the valley of decision. The sun and the moon shall be darkened, and the stars shall withdraw their shining. The LORD also shall roar out of Zion, and utter his voice from Jerusalem; and the heavens and the earth shall shake: but the LORD will be the hope of his people, and the strength of the children of Israel.
For I will gather all nations against Jerusalem to battle; and the city shall be taken, and the houses rifled, and the women ravished; and half of the city shall go forth into captivity, and the residue of the people shall not be cut off from the city. Then shall the LORD go forth, and fight against those nations, as when he fought in the day of battle. And his feet shall stand in that day upon the mount of Olives, which is before Jerusalem on the east, and the mount of Olives shall cleave in the midst thereof toward the east and toward the west, and there shall be a very great valley; and half of the mountain shall remove toward the north, and half of it toward the south.
From the books
This answer is drawn from these passages in the Mary Banks Faith Library.