In order to understand if the Bible teaches different levels of grace it is helpful to first define what it is.
Grace is found 170 times in the King James Version Bible. The vast majority of the time its Old Testament occurrences are derived from the Hebrew khane (Strong's Concordance #H2580) which means favor, graciousness or grace. In the KJV New Testament, it is derived all but once from the Greek charis (Strong's #G5485).
Grace is the disposition of undeserved generosity, kindness and favor that a giver has when giving a gift to another person (the receiver). When the Apostle Paul discuss it is his letters, he describes it as part of God's character whereby he gives people mercy and makes Christians righteous (i.e., "justification").
God offers his grace even though those who receive it cannot earn it based on good works.
Paul wrote in Titus 2, verse 11 that "the grace of God, which brings salvation for all men, has appeared." The Father loves all humanity, as this text shows, including all unfaithful and unrepentant sinners, but his favor is only effective in its results in those who believe. Notice in this light John 3:16, which states those who have faith in God will not die but receive eternal life. Unbelievers do not benefit from God's special favor until they believe and repent.
Favor toward believers
In one sense, all Christians have received a certain amount or "level" of the Eternal's grace since they needed him to possess the willingness to allow them to be saved despite all of their sins. Consider what Paul says in Romans 5.
Moreover, the law entered (was revealed to Israel after Adam's sin), so that transgression might abound (since people knew the law more, so they were more responsible and thus guilty when disobeying it); but where sin abounded, the grace of God did superabound (Romans 5:20).
However, past the basic amount needed for salvation, does God give his kindness in varying amounts or levels to those who are called and converted? The quick answer is "yes." Notice in particular what is written in the book of 2Peter.
Rather, be growing in the grace and the knowledge of our Lord and Savior Jesus Christ (2Peter 3:18).
Now, how can all (saved) Christians have the same level of favor, yet be implored by Peter to "grow" in it? Peter obviously does not mean, as Paul warned against, that we sin so then we can receive more of it (Romans 6:1)!
Grace has another effect on Christians besides gaining the basic gift of salvation after having the faith to receive the benefits of Jesus' sacrifice.
Believers can and do have differing levels and kinds of spiritual gifts and powers because of His will towards us. Paul was made an apostle by it (Ephesians 3:7 - 8). More generally, he later wrote that although there was one body (the church) and one Spirit, each person has received something different from the Eternal (Ephesians 4:7, 11).
Believers in the first century who attempt to be reconciled to God by becoming circumcised, and therefore had to keep the whole Old Testament law, are said to have fallen from His grace. Their loss would make them different from those who still had it.
Again, I am personally testifying to every man who is being circumcised that he is a debtor to do the whole law. You who are attempting to be justified by works of law . . . You have fallen from grace! (Galatians 5: 3 - 4)
Different levels of grace exist since non-believers do not benefit from having any from God, but true believers do. Even among those converted, varying amounts of it are bestowed in the form of the differing gifts of service he gives for the benefit of all.