What do we know about the “super apostles” (ESV) in Second Corinthians?

This phrase is used by Paul in two places—both in Second Corinthians (occurring in the NIV as well as the ESV). The first is in chapter 11 where he writes, But I do not think I am in the least inferior to those “super-apostles”—Paul, II Corinthians 11:5. The second time is in the very next chapter, where he writes, I have made a fool of myself, but you drove me to it. I ought to have been commended by you, for I am not in the least inferior to the “super-apostles,” even though I am nothing—Paul, II Corinthians 12:11. So what is he talking about? These people actually appear to have come in from outside the church. Most likely they were Jews who were trying to bring this particular church under the influence of the church in Jerusalem. Like the Judaizers, there was a sense that they were trying to get the Corinthians to conform to Jewish customs. However, they were not adamant about circumcision, nor were they particularly rigid in their doctrine, even apparently encouraging a disregard for strict rules. Their main influence, however, seems not to have come from the Jerusalem church (to which they falsely claimed approval) but from the prevalent Greek culture and philosophy.
More specifically, in the Greek, the translation of “super apostles” actually goes back to the phrase “very chiefest apostles,” literally meaning “more exceedingly sent one.” We get some hints of what that may mean in the context of some of Paul’s other comments. One of Paul’s most consistent worries in the churches he had planted was concerning the possible infiltration of false teachers. One of their methods, other than opposing Paul’s ideas, was to oppose Paul himself and his methods. If you can’t discredit the teaching, maybe try discrediting the teacher. Paul seems to suggest that that effort had already been underway in Corinth, as he says, just prior to his first mention of “super apostles,” But I am afraid that just as Eve was deceived by the serpent’s cunning, your minds may somehow be led astray from your sincere and pure devotion to Christ. For if someone comes to you and preaches a Jesus other than the Jesus we preached, or if you receive a different spirit from the one you received, or a different gospel from the one you accepted, you put up with it easily enough—Paul, II Corinthians 11:3-4. Then, after revealing the opposition as these so-called “super apostles,” he goes on to say, I may not be a trained speaker, but I do have knowledge. We have made this perfectly clear to you in every way—Paul, II Corinthians 11:6. One of the ways the “super apostles” may have presented themselves as superior to Paul was the possibility that they had received some level of formal rhetorical training that he lacked—creating a deficiency in his ability to speak well that he freely and readily admitted. Corinth was a city fully immersed in Greek culture and, as such, oratorical skills were prized. But, as he said in his first letter to Corinth, God chose the foolish things of the world to shame the wise; God chose the weak things of the world to shame the strong. He chose the lowly things of this world and the despised things—and the things that are not—to nullify the things that are, so that no one may boast before him. . . . When I came to you, brothers, I did not come with eloquence or superior wisdom as I proclaimed to you the testimony about God. For I resolved to know nothing while I was with you except Jesus Christ and him crucified. I came to you in weakness and fear, and with much trembling. My message and my preaching were not with wise and persuasive words, but with a demonstration of the Spirit’s power, so that your faith might not rest on men’s wisdom, but on God’s power. We do, however, speak a message of wisdom among the mature, but not the wisdom of this age or of the rulers of this age, who are coming to nothing—Paul, I Corinthians 1:27-29, 2:1-6. Paul was perfectly content being a preacher of the gospel versus a professional orator. However, in his knowledge of the gospel and its message, Paul was unsurpassed, particularly when considering the source of that knowledge. As he shared with the church at Galatia (ironically, another church plagued with false teachers), I want you to know, brothers, that the gospel I preached is not something that man made up. I did not receive it from any man, nor was I taught it; rather, I received it by revelation from Jesus Christ—Paul, Galatians 1:11-12. It also appears that they received payment for their efforts, while Paul continued to preach and teach free of charge—possibly underscoring their contention that they must be superior to Paul because their teachings had greater “value.” But in Paul’s eyes, these teachers were more clearly defined as agents of Satan who had been able to gain a hearing and some measure of influence in the church, and he pledges himself to stop them. As he makes so clear, I will keep on doing what I am doing in order to cut the ground from under those who want an opportunity to be considered equal with us in the things they boast about. For such men are false apostles, deceitful workmen, masquerading as apostles of Christ. And no wonder, for Satan himself masquerades as an angel of light. It is not surprising, then, if his servants also masquerade as servants of righteousness. Their end will be what their actions deserve—II Corinthians 11:12-15. Ironically, in Paul’s calling to preach the truth of the gospel of Jesus Christ, he had suffered much more than the “super apostles ever did (II Corinthians 11:23-27). In comparing his story with theirs, he is actually confirming his call and his path as superior to theirs—no matter what they might call themselves. And, of course, those apostles commissioned by Christ as His official spokesmen—including Paul—received power from Him to perform the same mighty acts as He did; as Paul confirmed, The things that mark an apostle—signs, wonders and miracles—were done among you with great perseverance—Paul, II Corinthians 12:12. No matter how the “super apostles” presented themselves, they would never be confirmed by the power of God through these signs. Instead, Paul confirmed them for what they were: pretenders of the apostolic mission and servants of Satan. While Paul might have mixed feelings about those opposed to him personally, he would never tolerate anyone opposed to him doctrinally. He would always fight that with everything he had.