Verses 1-18 of John chapter 1 can be read as a complete paragraph. At no point does John reveal Jesus' given name in this paragraph; rather, he calls him simply "the Word" and "the Light." In the section of the paragraph in which verse 16 finds itself, the mini-theme is grace.
πλήρης χάριτος καὶ ἀληθείας
plērēs charitos kai alētheias
"the Word became flesh . . . full of *grace* and truth," John tells us (v.14).
Moreover, as you point out in your question,
". . . of His [i.e., the Word's] fullness we have all received grace upon grace" (v.16),
". . . grace and truth were realized through Jesus Christ" (v.17).
A short but effective definition of grace (Gk. χάριτος/charitos) is "undeserved love in action."
Notice that John uses the words full (Gk. πλήρης/plērēs) and fullness (Gk. πληρώματος/plērōmatos) to describe that grace, which is his way of saying that Jesus is the source of an unlimited supply of grace. Moreover, grace precedes truth, according to verse 14. That order of grace before truth is significant, too, for according to the Law of Moses, the truth is we've all sinned by violating God's Law (see Romans 3:23, and James 2:10).
If when Jesus came onto the scene he had led with truth, all humankind would be proved guilty as lawbreakers. Jesus, however, led with grace and followed with truth. In other words, Jesus, in the fullness of God's grace, came not to judge and condemn the world
"but [so] that the world might be saved through Him" (John 3:17 NASB Updated).
What, then, does "grace upon grace mean"? The words full and fullness help answer the question. The phrase means there is grace, and then there is more grace; it is grace heaped upon grace; it is an inexhaustible supply of grace, available to us 24/7. It is all these things and more.
Put differently, there will never be a time when each of us will never not be in need of God's grace; or put positively, each of us will always be in need of God's grace (see, for example, Jesus' parable of the talents in Matthew 18:23 ff. for an excellent illustration of grace demonstrated and grace withheld).
Some believers look back on their conversion experience as being the pinnacle of their experience of God's grace in their lives, and that is perfectly appropriate and commendable. As John Newton put it in his famous hymn, "Amazing Grace,"
Amazing grace! How sweet the sound
That saved a wretch like me!
I once was lost, but now am found;
Was blind, but now I see.
’Twas grace that taught my heart to fear,
And grace my fears relieved;
How precious did that grace appear
The hour I first believed.
As we mature spiritually, however, many Christians find their experience of God's grace becoming even more precious to them, and they begin to see they will never "outgrow" their need for God's grace. In other words, there will never come a time in their Christian experience when they will cease being "sinners saved by grace."
That, then, is what "grace upon grace" means, at least in part. There is the precious grace we experience when we are born again and have
"passed out of death into life" (John 5:24).
There is, moreover, the precious grace we experience as we
"grow in the grace and knowledge of our Lord and Savior Jesus Christ" (2 Peter 3:18).
That growth in grace involves not only a deeper appreciation for God's grace in our lives, but also an increase in our modeling of that grace in the lives of others. As John Newton expressed this truth in the third verse of "Amazing Grace":
Through many dangers, toils and snares,
I have already come;
’Tis grace hath brought me safe thus far,
And grace will lead me home.
In conclusion, there will never come a time in the lives of believers in Jesus Christ when they will never be in need of a fresh infusion of God's grace. In fact, our experience of fullness of life in Christ can fairly be described as grace upon grace.