The Bible does not mention the exact word "purgatory," but instead it makes reference to a place.
To claim that it does not exist because of this is a cop out.
You might as well even deny that there is a book called the Bible because no such name is found in the inspired writings.
You also might as well deny the Trinity, Incarnation, etc... because these words are not found in the Bible.
The name does not make the place; the place must exist first, then we give it a name. We call this place "purgatory" because it means "a cleansing place." Therein souls are purged from the small stains of sin, which prevent their immediate entrance into Heaven.
In the Old Testament
The first mention of Purgatory in the Bible is in 2 Maccabees 12:46: "Thus he made atonement for the dead that they might be freed from sin."
Granted 2 Maccabees may not be in Protestant Bibles, but even if it cannot be used doctrinally then it at least has to have some historical worth. In it we can see what the pre-Christian community believed.
In Chapter 12 we can see Scriptural proof for Purgatory and evidence that the Jews had sacrifices offered for those of their brothers who had lost their lives in battle. That the Jews prayed for the dead shows that they believed in a place where they could be helped (now called Purgatory) and that the prayers of their living brothers and sisters could help them in that place.
These words in the book of Maccabees had so clearly favored the Catholic custom, that the whole book was removed from the Protestant Bible. Unfortunately for them, even if the book was not inspired, it still tells us of the practice of God's chosen people.
In the New Testament
In Matthew 5:26 Christ is condemning sin and speaks of liberation only after expiation. "Amen, I say to you, you will not be released until you have paid the last penny." Now we know that no last penny needs to be paid in Heaven and from Hell there is no liberation at all; hence the reference must apply to a third place.
Matthew 12:32 says, "And whoever speaks a word against the Son of Man will be forgiven; but whoever speaks against the holy Spirit will not be forgiven, either in this age or in the age to come."
The same person as in the previously mentioned verse, Matthew, speaks of sin against the Holy Spirit. The implication is that some sins can be forgiven in the world to come. But not in Hell from which there is no liberation; nor in Heaven because nothing imperfect can enter it as we see in the next part. Any remission of sin cannot occur in either of these places because they are a final destination unlike purgatory.
Revelation 21:27: "...but nothing unclean will enter it, nor anyone who does abominable things or tells lies." The place that is to be entered (the place to which this passage refers) is heaven (read the stuff around it for context).
The Bible clearly implies a place for temporary punishment after we die in the many passages which tell that God will reward or punish according to man's works.
What if purgatory really doesn't exist?
Say that there is no such thing as purgatory. What becomes of us? The Bible declares that nothing impurified can enter Heaven, but yet a careless word can defile the soul (Matthew 12:36); if there is no place of temporary punishment, the one guilty would be damned to Hell!
Who would be saved? Those who teach against purgatory teach an unreasonable doctrine.
So, why do non-Catholics reject a teaching so full of consolation? My guess is that they want to believe that the merits of Christ applied to the sinner who trusts in Him, will remove all sin; hence the believer will go at once to Heaven (also known as the belief called Sola Fide or faith alone).
Nowhere in the Bible does it say faith alone. This is un-Scriptural, since Christ tells us that to enter into life we must keep the commandments, hear the Church, do the will of His Father and much more with faith. Yes, actions plus faith.