If the Bible were merely an ordinary book, then its contents would be as exhaustible as any other book. However, the Bible declares itself to be anything but an ordinary book. Specifically, it describes itself as "living and active".
For the word of God is living and active, sharper than any two-edged
sword, piercing to the division of soul and of spirit, of joints and
of marrow, and discerning the thoughts and intentions of the heart.
Hebrews 4:12 ESV
It is the Spirit who gives life; the flesh is no help at all. The
words that I have spoken to you are spirit and life. John 6:63 ESV
Furthermore, a principle in Luke 6:45 is that a person's words reveals the heart of that person.
The good person out of the good treasure of his heart produces good,
and the evil person out of his evil treasure produces evil, for out of
the abundance of the heart his mouth speaks.
If, then, the words of God reveal the very heart of God Himself, then the Word of God is, in fact, inexhaustible, as God Himself is infinite.
Isaiah also speaks of how God's understanding is unsearchable:
Have you not known? Have you not heard? The Lord is the everlasting
God, the Creator of the ends of the earth. He does not faint or grow
weary; his understanding is unsearchable. Isaiah 40:28 ESV
So, from a Biblical perspective, the Bible, as the very words of God which reveal the nature of God, is, as a result, inexhaustible.
Since God is infinite, it will take an eternity to get to know Him fully. Fortunately, He has promised to give us exactly that amount of time to be with Him in Heaven.
As a side note, however, it is likely only inexhaustible for people who are actually Christians. Many believers attest to the fact that prior to their conversion to Christianity, the Bible was a confusing, lifeless, incomprehensible book, yet it came alive to them after their conversion. Paul hints at this phenomenon:
For the word of the cross is folly to those who are perishing, but to
us who are being saved it is the power of God. 1 Corinthians 1:18 ESV