Imam Ali (as) asked, "If you knew the meaning of the words of the call to prayer, you would laugh little and weep much. The statement "Allah is the Greatest" has many meanings.
When the muezzin says "Allahu Akbar" it refers to His Eternity, His Sempiternity, His Infinity, His Knowledge, His Power, His Omnipotence, His Forbearance, His Munificence, His Generosity, His Bounty, and His Pride. When the muezzin says "Allahu Akbar" he proclaims that the Creation and the Command belong to Allah. He proclaims that creation came about as a result of His Will. He proclaims that He is the First of the First and He has always existed, and that He is the Last of the Last as He will always exist. He proclaims that He is the Manifest, who cannot be perceived, and that He is the Hidden, who is not subject to limitations of any sort. He proclaims that He is the Ever-Lasting, and that everything but Him will perish.
The second meaning of "Allahu Akbar" refers to the fact that He is All-Knowing, the All-Informed, who knows everything that has occurred and everything that will occur, even before it occurs.
The third meaning of "Allahu Akbar" is that He is All-Powerful over everything. His power extends over everything. He is powerful due to His Power, and the Potent One over His Creation. He is Powerful in Essence, His Power embraces all things. When He decrees a thing, He only says to it, "Be," and it is.
The fourth "Allahu Akbar" refers to His Forbearance and Generosity. He forbears as if He were unaware of the sins, pardons as if He did not see the sin, and covers with mercy as if He had not been disobeyd. He does not hasten punishment out of His Generosity, Forgiveness, and Forbearance.
Another meaning of "Allahu Akbar" is that He is Generous, Ample-Giving, and the Most Munificent. The other meaning of "Allahu Akbar" is to negate His Shape, and that Allah is Greater than any of His Attributes. Verily, those who describe Allah describe him on the basis of their own ability and not according to His Might and Majesty. Exalted is Allah, the Elevated, the Great, from what the attributes perceive as His Attributes.
Another meaning of "Allahu Akbar" refers to the fact that Allah is the Most High and Exalted. He is Self-Sufficient from His Servants. He is not in the need of the deeds of His Creation.
[When in the end of Adhaan when muezzin recite final two Allahu Akbar, Imam Ali says...]
As for his statement "Allah is the Greatest" [Allahu Akbar] it means: Allah's Kindness towards His Creation is greater than anyone can possibly conceive. It refers to His Kindness towards His Servants through answering their prayers; His Kindness towards His Servants who obey Him, and who obey His Legal Guardians; His Kindness towards His Servants who servers Him, who preoccupies himself with Him and His Remembrance, who loves Him, and who befriend Him; His Kindness towards His Servants who finds tranquility in Him, who trusts Him, who fears Him, who puts his hope in Him, who yearns for Him, who accepts His Commands and Decrees, and who is pleased with Him.
The second time that [Allahu Akbar] is said it means: Allah's Kindness towards His Creation is so great that it cannot possibly be conceived. It refers to His Kindness towards His Friends, His Punishment of His Enemies, the extent of His Pardoning, His Forgiveness, and His Bounty towards those who answer His Call, and the Call of His Messenger. It also refers to the extent of His Punishment and His Humiliation of those who deny Him and who reject Him." from Kitab al-Tawheed.