If you want to make a Calvinist roll their eyes, cite John 3:16. If you want to put them on their back foot, insist they address it. John 3:16 is one of the most cliche responses to Calvinism, in particular to Limited Atonement. As John 3:16 tells us "...God so loved the world that He gave His only begotten Son, that whoever believes in Him should not perish but have everlasting life." (Emphasis added), one can only logically conclude that God's love extends beyond those who believe. Thus, even though God knew ahead of time who would believe, it was at least possible for the rest to believe and be saved.
Further evidence that Christ made propitiation for the sins of the whole world comes from 1 John 2:2: "And He Himself is the propitiation for our sins, and not for ours only but also for the whole world." (Emphasis added). The problem this presents for Calvinists is that it clearly and explicitly tells us that God loves, and died for, the whole world, not only for believers.
Calvinist responses to this, as far as I have seen, go only as far as a denial of the most obvious interpretation, but do not provide an alternative interpretation, nor a reason God used this particular phrase. If the Calvinists were correct, we would expect the verses to read something like "for God so loved the elect that He gave His only begotten Son, that when they come to believe in Him, they will not perish, but receive eternal life", and "He Himself is the propitiation for our sins, and not for ours only, but also for the whole elect". Instead, God opted to use the term "world".
This is where Calvinists point out that "world" does not necessarily mean all people. Luke 2:1, for example, does not show Caesar Augustus trying to register anyone outside of his jurisdiction. And that's entirely true. It is legitimate to understand "...all the world...", in this case, as referring only to a select group of people. The problem is, we cannot use this particular interpretation without ruling out 90% of the modern Church. And in fact, there is no definition of the word "world", much less a reason for using it without some form of context or qualification, that would cover all Christians without also covering all people.
This is because God actually wants, and will have, people from every world saved. Every tribe, every tongue, every nation, every people, will be represented in the Kingdom of Heaven (Revelation 5:9; 14:6). This is not surprising, as Paul tells us God wants all men to be saved and come to the knowledge of the truth (1 Timothy 2:3-4).
The standard Calvinist objection would be that unless we want to preach Universalism, the standard interpretation of the word "world" simply cannot cover unbelievers either. The alternative is God failed. There are several problems with this, starting with Isaiah 55:9: "“For as the heavens are higher than the earth, So are My ways higher than your ways, And My thoughts than your thoughts." What this means is it is not for us to decide when scripture is and isn't true, even if we don't like the implications. If Jesus tells us God loves the world, then God loves the world. If John tells us Christ made propitiation for the sins of the whole world, then He made propitiation for the sins of the whole world. If Paul tells us God wants all men to be saved, God wants all men to be saved. It is not our place, nor even our right, to tell Him "well then you failed". A statement I hope every Calvinist will agree with: "The Bible does not need to be re-written, only re-read".
For this reason, we can reject the Calvinist idea that when God says He wants all men to be saved, He actually meant to say "all kinds of men". In fact, that wouldn't even fit with the context of the verse: "Therefore I exhort first of all that supplications, prayers, intercessions, and giving of thanks be made for all men, for kings and all who are in authority, that we may lead a quiet and peaceable life in all godliness and reverence. For this is good and acceptable in the sight of God our Savior, who desires all men to be saved and to come to the knowledge of the truth."
In this passage, we are told to make prayers, supplications, intercessions, and thanksgivings, for all men, for kings, and for all who are in authority. This is good in God's sight because He wants all men to be saved. The "alls" correspond to each other here. So unless the Calvinist wants us to make some very confusing assessments on which kings it is appropriate for us to pray for, then God wants all men, kings, and authorities, to be saved. Their own method of interpretation refutes the interpretation they came up with the method to defend!
So, clearly God wants all men to be saved, but did He therefore fail? Well, pay attention to John 3:16. God loved who? The world. Who will not perish? Whoever believes. Was that goal accomplished? Absolutely. All who believe will not perish, but will have eternal life.
This is actually seen in some of the ways Jesus describes Himself. In particular, Jesus describes Himself as "the door of the sheep", and that anyone who enters by Him will be saved (John 10:7-9). The thing about doors is as long as they are open, there is no limit on who can walk through them. If the door is open, Jesus succeeded, even if only a few go through.
But is that really all it is? Did Jesus really make propitiation for the sins of the unsaved? Consider what Peter says in his second epistle: "But there were also false prophets among the people, even as there will be false teachers among you, who will secretly bring in destructive heresies, even denying the Lord who bought them, and bring on themselves swift destruction. And many will follow their destructive ways, because of whom the way of truth will be blasphemed. By covetousness they will exploit you with deceptive words; for a long time their judgment has not been idle, and their destruction does not slumber." (2 Peter 2:1-3, emphasis added).
Although Jude does not specify the Lord bought these men, he does speak of those who "...crept in unnoticed, who long ago were marked out for this condemnation, ungodly men, who turn the grace of our God into lewdness and deny the only Lord God and our Lord Jesus Christ." (Jude 1:4). Comparing both chapters, it seems both Apostles are describing the same types of people.
There can be no doubt that these people were unsaved. Both deny the Lord, and both are destroyed and condemned. Yet, the Lord... "bought" them. How is this possible? How can the Lord "buy" people who already belong to Him? Only in one way: By redeeming them with the blood of Christ. This alone makes sense of why Christ not only bought them, but also destroys them in spite of having done so.
But if Christ forgave the sins of these people, how can they not be forgiven in the end? To this, I say go to the parable of the unmerciful servant (Matthew 18:21-35). In this parable, the servant receives genuine mercy, and yet, being himself devoid of mercy, his debt is returned to him. From this, we see that the forgiven, under certain circumstances, can be unforgiven.
This even makes sense of why God's love always seems to be active in scripture, but He seems to be begrudgingly wrathful. He takes no pleasure in the death of the wicked (Ezekiel 18:32; 33:11). He is patient because He is not willing that any should perish (2 Peter 3:9). He endures with much longsuffering the vessels of wrath (Romans 9:22). Even their condemnation and reprobation is because they do not believe (John 3:18; Romans 1:28).
With all of this in mind, we can afford to stop playing silly games by making up new definitions of words, or worse, taking them out or putting them into scripture, and just let scripture say what it says. If scripture says God loved the world, God loved the world. If Christ made propitiation for the sins of the whole world, He made propitiation for the sins of the whole world. If God wants all men to be saved, God wants all men to be saved. It is 100% acceptable, and indeed is absolutely necessary, for us to just believe that, rather than deciding it doesn't bode well for a 17th century pastor's strange beliefs and so we must change it. God is wiser than John Calvin, and so we must conform Calvin's words to God's.