...not clarifications or particular interpretations that Book of Mormon might put on things that are in the Bible, but rather doctrines Mormons believe in that have no Biblical basis
That can be a difficult distinction.
For example, the Book of Mormon adds the belief that Adam and Eve could not have children before the Fall.
And now, behold, if Adam had not transgressed he would not have fallen, but he would have remained in the garden of Eden. And all things which were created must have remained in the same state in which they were after they were created...
And they would have had no children; wherefore they would have remained in a state of innocence, having no joy, for they knew no misery; doing no good, for they knew no sin.
2 Nephi 2:22-23
This concept (no fertility until after the Fall) is not taught in the Bible.
So is this a "new" doctrine taught in the Book of Mormon and not the Bible, or is it considered only a "clarification" of an existing Biblical doctrine?
In any case, there are many examples, so this list is not exhaustive:
There can (should) be more temples than just the Temple at Jerusalem:
And I, Nephi, did build a temple; and I did construct it after the manner of the temple of Solomon...and the workmanship thereof was exceedingly fine.
2 Nephi 5:16
The temples of the LDS church have similarities with Solomon's temples, e.g. a basin of water on the backs of twelve oxen. This is a striking difference, as other Christian denominations do not have such temples.
Jesus existed in the form of a man even before his birth.
And the Lord said unto him: Because of thy faith thou hast seen that I shall take upon me flesh and blood...
Behold, this body, which ye now behold, is the body of my spirit; and man have I created after the body of my spirit; and even as I appear unto thee to be in the spirit will I appear unto my people in the flesh.
Parts of the Bible have been altered by men.
And the angel of the Lord said unto me: Thou hast beheld that the book proceeded forth from the mouth of a Jew; and when it proceeded forth from the mouth of a Jew it contained the fulness of the gospel of the Lord, of whom the twelve apostles bear record...
Wherefore, thou seest that after the book hath gone forth through the hands of the great and abominable church, that there are many plain and precious things taken away from the book, which is the book of the Lamb of God.
...because of these things which are taken away out of the gospel of the Lamb, an exceedingly great many do stumble...
1 Nephi 13:24,28-29
A prophet named Joseph, whose father is named Joseph, will (has) come as Moses to convince Israel of God's word.
Yea, Joseph [son of Jacob] truly said: Thus saith the Lord unto me: A choice seer will I raise up out of the fruit of thy loins...
And he shall be great like unto Moses, whom I have said I would raise up unto you, to deliver my people, O house of Israel.
...and not to the bringing forth my word only, saith the Lord, but to the convincing them of my word, which shall have already gone forth among them.
And his name shall be called after me; and it shall be after the name of his father...
2 Nephi 3:7,9,11,15
This is a reference to Joseph Smith Jr., the first prophet of the Church of Jesus Christ of Latter-day Saints.
A portion of scattered Israel is in the Americas:
And now, behold, I would speak somewhat unto the remnant of this people who are spared...
Know ye that ye are of the house of Israel.
And ye will also know that ye are a remnant of the seed of Jacob; therefore ye are numbered among the people of the first covenant...
These people are known in the Book of Mormon as the Lamanites. For this reason, the early LDS Church made efforts to convert the Native Americans (and in modern times, the LDS Church has grown quickly in Central and South America).
God will reveal additional ancient records of the Lost Tribes of Israel, beyond the Bible and Book of Mormon.
For behold, I shall speak unto the Jews and they shall write it; and I shall also speak unto the Nephites and they shall write it; and I shall also speak unto the other tribes of the house of Israel, which I have led away, and they shall write it; and I shall also speak unto all nations of the earth and they shall write it.
And it shall come to pass that the Jews shall have the words of the Nephites, and the Nephites shall have the words of the Jews; and the Nephites and the Jews shall have the words of the lost tribes of Israel; and the lost tribes of Israel shall have the words of the Nephites and the Jews.
2 Nephi 29:12-13
In regards to Kolob, that name is not in the Bible or Book of Mormon. It is only mentioned in the Pearl of Great Price, and in the Doctrine and Covenants. It is the star nearest to God.
And while that particular name is never mentioned in either the Bible or the Book of Mormon, both teach that God's throne is the heavens.
But verily, verily, I say unto you, swear not at all; neither by heaven, for it is God's throne;
3 Nephi 12:34, also Matthew 5:34
which, considering no one knows where Kolob is, is about as specific :)