Many in the Christian community claim that Jesus’ promise to the disciples that they would do greater things than Him means they can do even greater miracles than Jesus did, but what did Jesus mean by saying “greater things?”
When Jesus began His earthly ministry, the people were in awe of Him, and “many of the people believed in him. They said, “When the Christ appears, will he do more signs than this man has done” (John 7:31)? The Samaritan woman at the well knew He was the Messiah, saying to the people in her village “Come, see a man who told me all that I ever did. Can this be the Christ” (John 4:29)? The Jewish religious leaders, because they envied Jesus’ following, asked Him “Then what sign do you do, that we may see and believe you? What work do you perform” (John 6:30), but Jesus’ customary answer was, “This generation is an evil generation. It seeks for a sign, but no sign will be given to it except the sign of Jonah” (Luke 11:29). God is still doing miracles today, I have no doubt. He is God, but miracles always point to the glory of God and never draw attention to the person whom God worked through. When you read the gospels, you might notice that when the crowds began to grow large, and this was often the case, He would speak very clearly about what was required, once saying, “If anyone would come after me, let him deny himself and take up his cross and follow me” (Matt 16:24). Telling the crowds what following Him would cost them, often thinned out the crowd a bit. Yes, Jesus did many great things, but they were done to validate that He was God and God was working through the Son. Jesus never used miracles to draw attention to Himself, but always gave the glory to the Father. When Jesus healed someone, He told them to tell no one about it (Mark 7:36). Jesus made the lame to walk, the mute to speak, the blind to see, the leper to be cleansed, and the dead to life! These are really great things, so what exactly did Jesus mean when He said, “whoever believes in me will also do the works that I do; and greater works than these will he do, because I am going to the Father” (John 14:12).
During the early years of the church, God was working wonders through the apostle’s hands, and some “even carried out the sick into the streets and laid them on cots and mats, that as Peter came by at least his shadow might fall on some of them” (Acts 5:15), and “The people also gathered from the towns around Jerusalem, bringing the sick and those afflicted with unclean spirits, and they were all healed” (Acts 5:16). Jesus had told them before going back to the Father that there would be many signs that would accompany them (Mark 16:17), and if you read the first part of the Book of Acts, you can see that God was working so powerfully through the apostles. The word got out, so that “even handkerchiefs and aprons that had touched him were taken to the sick, and the diseases and evil spirits left them” (Acts 19:12). These were things that Jesus did, so how could Jesus tell them that they’d do “greater works than these” (John 14:27)? What greater works did the apostles ever do that superseded the things that Jesus did? Was it healing? Jesus did that. What is making the lame to walk? Jesus did that, so was Jesus speaking only to the apostles or also to us? And, what greater things than these can we do today than Jesus did?
What is the greatest thing that any believer could do? I believe it is to share the gospel with someone so that they might be saved. Naturally, God does the saving, and it’s not our responsibility to save anyone. It is their response to His ability, but it is our responsibility to share the good news with lost people in the hopes that God grants them repentance (2 Tim 2:25). If God uses us to heal someone by praying for them, that’s great…all glory to God, but to have someone healed while still rejecting the gospel means they’ll only live a healthier life on their way to hell. What is greater than rescuing a person who would otherwise perish apart from Christ? Does healing a sickness or disease equal eternal life in Christ? No way! That’s greater than any physical healing, isn’t it? Even self-proclaimed faith healers die…but Jesus said, “Whoever believes in me, though he die, yet shall he live” (John 11:25b). You can be healed by a miracle and still die in your sins, but when a person puts their trust in Christ, they can endure in this life, even if they’re never healed. They know that not even death can separate them from God (Rom 8:38-39). They also understand that God uses suffering for His glory and our best. We are refined by being passed through the flames, yet not tossed into the fire. When someone told me they wanted to teach me how to speak in tongues, I asked, “I would rather you teach me how to use the Word of God to save someone.” The tongue, being among the smallest members of the body, can bring about the best…like in unashamedly proclaiming the gospel, which has the power of God to save (Rom 1:18; 1 Cor 1:18).
For those who are running around the world claiming to do greater works than Jesus, I would ask, “Do you really believe that you have exceeded what Jesus did here on earth?” If Jesus only meant that they would do greater miracles than Him, I think it does injustice to the context of His words. Jesus was commissioning them to make disciples of all people. The scope of Jesus’ earthly ministry was basically centered near Jerusalem, but the apostles would spread out far and wide, and leave behind the Word of God (in the gospels and letters). Bartholomew would go to Asia, Matthew to Ethiopia, and Thomas to India. All but John would die a martyr’s death. The gospels would go on to be published and spread worldwide, so they are actually still doing great things, and together, as they spread out, they reached further than Jesus’ earthly ministry ever could, being limited by space, time, and distance.
I notice that the miracles began to cease in the Book of Acts, as the church began to be established. I think that’s indicative of the fact that the early church, which did not have the New Testament, was validated by the signs and wonders as evidence that God was working in these disciples. The apostles are still doing “greater things” by reaching all nations with the Word of God because much of the New Testament comes from the original apostle’s writings, but today we’re able to reach even greater audiences than they could. We have greater technology and can reach more people than Jesus or the apostles could, but, if they mean that they are doing greater miracles than Jesus, they don’t understand the context of Jesus sayings. Besides, I believe that the greatest miracle for us was our conversion. A person once dead in their sins and held captive by the enemy is now quickened to life (Eph 2:1-2) through trusting in Jesus Christ. God can and still is doing miracles, I am sure, but what greater thing is there than to be used by God to rescue the perishing from the gates of hell, and bring them into the kingdom by the Word of God and the Spirit of God, and all for the glory of God.
Here is some related reading for you: Does God Still Work Miracles Today? 
Resource – Scripture quotations are from The Holy Bible, English Standard Version® (ESV®), Crossway Bibles. (2007). ESV: Study Bible : English standard version. Wheaton, Ill: Crossway Bibles. Used by permission. All rights reserved.