General view -- outside perspective
It is commonplace for Christians to conclude that theological views they do not personally believe are theological views without a Biblical basis. This would be true whether the rejected theological views were held by Latter-day Saints or non-Latter-day Saints.
The formal argument representing this implicit belief is as follows:
P1: If it had a Biblical basis, I would believe it
P2: I don't believe it
C1: It does not have a Biblical basis.
(I readily acknowledge there are some Christians willing to admit Biblical ambiguity on at least some matters; such individuals graciously exhibit charity to those who disagree with them, and would not accept P1 above)
General view -- inside perspective
To a Latter-day Saint, rejecting a theological view because it is found in other sacred texts, but not clearly found in the Bible, would be like asking a Protestant to reject theological views which are found in the epistles of Paul but not in the 4 Gospels.
(one could also imagine a stipulation that all New Testament doctrines must have an Old Testament basis in order to be accepted--there was real, historical debate on that matter; or that the only statements in the Gospels that should be accepted as true are those found in whichever Gospel was written first)
Sometimes those who are less-favorably-disposed towards The Church of Jesus Christ of Latter-day Saints consider it a accomplishment to elicit Latter-day Saint acknowledgement that they believe things that are not found in the Bible. Since all of us believe things that are not found in the Bible, this isn't much of an accomplishment. Furthermore, it fundamentally misunderstands Latter-day Saint beliefs about scripture.
The Church of Jesus Christ of Latter-day Saints accepts the Bible, the Book of Mormon, the Doctrine & Covenants, and the Pearl of Great Price as canonical scripture (going forward I will refer to the latter 3 as "the texts of the Restoration"). There is no requirement that a statement must appear in all 4 in order to be true.
A number of things Latter-day Saints believe are found only in the Bible--these principles are not rejected because they are not found in the texts of the Restoration. Likewise, principals taught only in the texts of the Restoration are not rejected because they are absent from the Bible.
Let us consider the following sets (further subsets could be created but are unnecessary to make the point):
- Set A: teachings found only in the Bible
- Set B: teachings found only in the texts of the Restoration
- Set C: teachings found in both the Bible and the texts of the Restoration
Anything in set B or set C would be examples fitting the OP's criteria: What are examples of Latter-day Saint doctrines that are NOT completely based on the Bible?
For sake of space, I will provide just 2 examples for each set.
The Nativity accounts of Jesus (including choirs of angels, shepherds, magi, etc.); Jesus' testimony to Mary in John 20:17.
The infinite & eternal nature of Jesus' atoning sacrifice, as taught in Alma 34:8-16; Moroni's promise of the revelation of truth, as taught in Moroni 10:3-7.
We'll start with an uncontroversial example: Jesus rose bodily from the dead--this is clearly taught in the Bible and the texts of the Restoration.
Many teachings that Latter-day Saints believe exist in Set C are considered controversial by other Christian faiths. For example, the pre-mortal existence of humanity. This is undoubtedly taught in Alma 13 & Abraham 3 (texts of the Restoration); however, Latter-day Saints also see evidence of this doctrine in the Bible (e.g. Job 38:7, Jeremiah 1:5, Eccl. 12:7, John 9:2, 2 Tim. 1:9, etc.). In this case, a doctrine taught in the Bible is given clarity and additional concrete detail via texts of the Restoration.
The Relevant Latter-day Saint Theology
We believe all that God has revealed, all that He does now reveal, and we believe that He will yet reveal many great and important things pertaining to the Kingdom of God. (Articles of Faith 1:9)
We reject the claim that God's silence on a matter in the past (or humanity's loss of something He said in the past) imposes any restriction on God's ability to speak in the future.
Disclaimer: these views are the product of my own study and do not constitute official statements by The Church of Jesus Christ of Latter-day Saints