Jeff asks, "What does 'being filled with the fullness of God' from Ephesians 3 mean?"
In that section of Ephesians, Paul is praying for the church. Among other things, he prays that they…
may have strength to comprehend with all the saints what is the breadth and length and height and depth, and to know the love of Christ that surpasses knowledge, that you may be filled with all the fullness of God (Ephesians 3:18–19 ESV).
Paul desired that his readers would know and experience the love of Christ fully. But as you implied in your question, Paul acknowledged that full comprehension of that love is impossible because it is greater than mortals can imagine.
Paul is praying for the ideal outcome: that we would know and love God fully. It’s similar to the desire Jesus expressed when he commanded us to “be perfect, as your heavenly Father is perfect” in Matthew 5:48. We can’t be perfect, or know and love God fully on this side of heaven. But that is the ideal for which we pray and strive.
The four words that Paul used—width, length, height, and depth—indicate the vastness of what it means to comprehend God. The reality is, no matter how much we know of the love of Christ, there is always more to know.
The ultimate goal of Paul's request was that we might be so full of the knowledge of Christ's love and appreciation for God that we will allow Christ to control us fully.