Is there scripture in the Bible referring to the denial of the Spirit a specified number of times before the Spirit ceases dealing with a person?
I could find no scripture stating or implying that the Spirit of God will give up on anyone after they deny Him a certain number of times. That said, there are some scriptures that may apply to this question.
God expects us to forgive an unlimited number of times. In Matthew 18:21-22 Jesus set a number of times to forgive at "seventy times seven," or 490. But once a person is forgiven the first time, if he is truly forgiven, then the numbering starts again at one, so to speak, so this is an extremely high, if not infinite, number. In Luke 17:4 the number was set as at least seven times a day. If God expects us to be that forgiving, then surely His Spirit is even more forgiving, and would not put a number on forgiveness.
On the other hand, it is possible for those who have received the Spirit of God to "quench" that spirit (1 Thes 5:19). The writer to the Hebrews spoke of those who sin by choice after having obeyed the truth as "doing despite to the Spirit." "For if we sin wilfully after that we have received the knowledge of the truth, there remaineth no more sacrifice for sins, But a certain fearful looking for of judgment and fiery indignation, which shall devour the adversaries. He that despised Moses' law died without mercy under two or three witnesses: Of how much sorer punishment, suppose ye, shall he be thought worthy, who hath trodden under foot the Son of God, and hath counted the blood of the covenant, wherewith he was sanctified, an unholy thing, and hath done despite unto the Spirit of grace?" (Heb 10:26-29) He also says, "For it is impossible for those who were once enlightened, and have tasted of the heavenly gift, and were made partakers of the Holy Ghost, And have tasted the good word of God, and the powers of the world to come, If they shall fall away, to renew them again unto repentance; seeing they crucify to themselves the Son of God afresh, and put him to an open shame." (Heb 6:4-6) Note that all these passages are talking about those who have received the Spirit as a result of baptism for the forgiveness of sins, and then have chosen to turn their backs on God. The Spirit even here is willing to forgive any number of times, but it will be harder to convince the fallen individual of his need for salvation because he has rejected the arguments that convinced him in the first place.
The only limit I know of to God's willingness to forgive is death. After that our position before God appears to be sealed. Before that, He is "not willing that any should perish, but that all should come to repentance." (2 Peter 3:9)