Many have said that the only thing that God the Father can not do is that He can not sin, or as the scripture says "can not lie"* Tit 1:2. Other than that, in terms of ability, of course God can do anything. After all, if God can not do things, Who can? Whoever this being is will have to be God instead.
The fact is there are many things that God the Father can not do, not because He is restrained by anything outside of Himself, but exactly the opposite, His very nature is His own restraint. Please ponder on these following examples of things God can not do, only because He is God.
(1) God the Father can not co-exist with; or look upon sin. Not only that; but any space or matter that He sanctifies for His own purpose can not be touched by sin. We know of such cases of people who did not heed this awful truth; and what the consequences were. The ark of the covenant in the land of the Philistines (1Sam 5), Uzzah (2Sam 6:6-9), Achan (Josh 7:1, 25-26), Acts 5:1-11), The church of Corinth (1Cor5:4-5), (1Cor 3:16-17).These are few examples that I can think of. In such cases God has to depart or strike.
(2) God the Father can not be humble; taking humility here to mean to not be completely wrapped around and absorbed in one's own self. God the Father can not give any reverence, submit to, fall in adoration with anything or anyone other than Himself. The Scriptures tell us that He loves to reside with the humble( Is 57:15 ), but that is different from being humble. In our human understanding, Humble people are those who associate with those of low status. How then does God loves to dwell with the humble, yet not be humble Himself. The answer is found in the fact that God is God, and we are not. We aught to be humble because we are created. To not be humble for a created being is a grievous sin, even if that being is created perfect ( Ez 28:12b-17) He has no reason to glory in self, but only in his creator. On the other hand if God glories in anything other that Himself, then this thing deserves to be God. How then can God take care of His creation if He is utterly wrapped around Himself? God can do that with perfect ease due to His infinite nature.
(3) God the Father can not be seen. The scriptures tell us that none had ever seen the Father, nor anyone can ever see Him, (1Tim 6:16), now and for eternity except for the persons of the Trinity. In (Is 6:2) we are told that the Seraphim have six wings, with two they cover their faces. They serve a being that they can never take a glimpse at, lest they be consumed.
(4) God the Father can not die, since he has no body nor is there any physical aspect to Him.
(5) GOD the Father can not deny Himself 2 Tim 2:13.
(6) God the Father can not change. This is a critical point because it is commonly understood to mean that God can not change the way He does things.** While it is true that God is unchangeable, we see clearly form His Word that He has many times changed the way He did things, not because His wisdom or knowledge were lacking in any way, but simply because He wants to reveal to His finite creation all of His attributes, and not focus on some without the others. In this we see how by focusing on some of God's attributes such as His mercy, love, longsuffering, etc...more than others. It robs Him of His glory, and of course the opposite is true as well.
These are few of the things I can think of that God can never do, you may add to that the fact that God can not do anything that is contrary to His nature. Example:God the Father is a spirit means He can not have a body, faithful means He can not promise and not fulfill, Just means he can not do injustice, love what is good and hates what is evil, etc...
But that does not mean that God the Father has not found means to declare His person to His creation. Here we find the necessity of the person of our Lord Jesus. He humbled Himself, took on human flesh, went under the power of death (Phil 2:7-8) , made Himself sin (2Cor 5:21), ate with and touched sinners (Matt 9:11), yet without sin ( Heb4:15). In these things although the Son Is equal to the Father In His essence(inner makeup,or deity) (Jon 14:9-11), Yet He may also say to the same disciples in the same chapter "The Father is greater than I " (Jon 14:28), also see (1Cor15:28). This is a mystery, how the Father, the Son, the Holy Spirit are one in essence, but at the same time distinct in their person. In that way we may say that the Son can do things the Father can not. It is not a matter of having more ability, but rather flexibility, because of His human nature.
The main point of this is to show the importance of actual sanctification in order to have communion with the Father through His Holy Spirit, esp point (1), or even to abide in Christ since He is now in His glorified state. You may say that there are two manifestations to Christ. The one who lived on earth 2000 years ago( this Christ is no more), and the one to come is described in ( Rev 1:13-18, 14:14-16, 19:11-16).
* Please note that God calls things that do not exist as if they do exist. ( Rom 4:17b), because of His infinite power and foreknowledge. So if God says that an army of hosts is camping outside, we better believe Him even if reason and our senses prove the opposite.
** In the story of Noah, God regretted that he destroyed the ancient world with flood, and promised that He will not ever destroy the world with flood again. If God had to do it all over again, He will still destroy the ancient world with flood, and promises not to do it again. Also in (1 Cor10:1-11), the children of Israel might have reasoned that if God is unchangeable, He can not destroy them after He redeemed them out of Egypt with a mighty arm, fed and cared for them in the wilderness. The answer to that resides in point (1).
