As the title asks, how many bilateral covenants occur in scripture between God and His creation (this does not include bilateral covenants between humans and humans, e.g., a marriage covenant between a man and a woman, or the covenant between Avraham and Avimelekh).
Although we often find various covenants spoken of in scripture such as the covenant with Adam, Noah, Abraham, Moses, etc. there are really only two covenants between God and man. The remaining covenants are merely modifications, progressions, or what might be called dispensations of the only two covenants.
The first covenant is the covenant of works established in the creation of Adam and Eve. By virtue of being created mankind was under natural duty to love and be absolutely obedient to God. If Adam were to fully obey God, during some period of testing, he would have obtained eternal life for humanity from works. That is to say that Adam was a federal head and representative of the covenant of works with respect to humanity. Upon Adam's sin or obedience, mankind would have been imputed with death and sin, or righteousness and life forever.
Upon the breaking of that first holy covenant, resulting in the death and moral fall of all mankind, a new covenant was required according to the love and mercy of God as well as his justice in being unable to forgive sin. The second and final covenant was the covenant of grace being promised through the seed of the woman, who was to crush the head of the serpent and restoring the creation to God.
These two single covenants run parallel throughout the redemptive history of scripture and never absolutely join as they are opposed to one another, in that one is of works and one is of grace. From Adam to the end of the world, man has and always will have the choice of remaining condemned under the covenant of works, or through faith in the promised seed, obtain new birth and eternal life under the covenant of grace. Before Christ, men believed in the new covenant as a promise, after Christ men believed it as a historical proclamation n the gospel.
That there was a long and steady degeneracy of the original knowledge of the covenant of works is evident in the progression of idolatry. At the same time that there was a slow developed progression of the knowledge of the covenant of grace until the climatic revelation of Jesus Christ and the full clarity of the gospel and law as proclaimed in the Epistles of the New Testament, is evident in the history of Israel and the gospel eventually extended to the gentile world. To properly understand these two covenants and to properly distinguish them from one another is the most significant work of theology.
Under the Old Testament, the knowledge of the original law and the promise seems to have degenerated greatly in the world until the time of the flood. In the calling of Abraham, God called a people to preserve the understanding of law as well as the future promise. This preservation by race preserved the knowledge and curse of the law by their commands, its severe punishments and their continuous national failure to obey it. This race also preserved the promise in its actual the lineage of the great Messiah to come, pointed to by their various ceremonial prefigurations of atonement. By re-starting humanity, in a sense, under Noah, purging the filth of the world, God modified the covenant of works and re-established the covenant of grace. By "killing the whole world' law and its curse on sinful man was revived. By putting up the rainbow high up into view, the promise was then renewed. These are not new covenants strictly speaking just renewals of the same two covenants. Through Abraham, both covenants were again further strengthened in separating a people unto holiness, condemning the world through law, while also promising the seed more explicitly along specific descendants.
In calling Moses to establish an outward form of religion, first this clearly re-enforced the covenant of works in its condemnation of sin and fearful capital punishments for the least offense. Again also the ceremonies and theocracy figuratively further established both the law and the covenant of grace. The ceremonies established law in enforcing the need of forgiveness, while establishing grace in prefiguring Christ. The theocracy established law in the commands of the King and his support of the law with its civil rules direct under God, yet also again prefigured the grace in the resurrection of Christ. Both covenants were made stronger in men's understanding through Israel, while yet being of opposed means, one of works the other of faith.
Having established both parallel covenants under one system of practice, the covenant of works under the Law and the covenant of grace to redeem from that Law, a greater light of these two covenants was provided for through the Mosaic arrangement. Furthermore, the Law it its condemnation and death, was now through figures and ceremony directed and officially pointed to unto a new end and new covenant, for all pointed to the work of the High Priest and the atoning sacrifices which was not so wonderfully arranged into clear picture before this time. The new covenant was to be the end of the old covenant and so the theocracy, ceremony and outwardly written laws under them. The end of the old law by the new is not a destruction but a superseding and fulfilling end. However the old law and its claims upon the guilt of the sinner is absolutely destroyed in Christ's death otherwise there is no salvation for anyone.
In the fullness of time, that is when the world and the Jewish church had both fully fallen into an absolute failure to find any means of saving themselves, Christ appeared. In his death, he destroyed the demands of the first covenant, ended the temporary figurative and theocratic elements of the dispensation of a physical church, provided the full accomplishment of the priestly works to ratify the new covenant, and rose and entered into glory to act as the new federal head and representative of a new humanity. By raising up as a King, Priest and Prophet, sending his Spirit into the world to continue his work until the end of the age, the first covenant was officially superseded and its claims on the conscience destroyed. The new covenant was absolutely ratified before the court of the Father and to the delight of angels and men. The Spirit now works upon the consciences of men, saving those that perish by proclaiming this gospel with certainty to the eternal salvation of any soul that will but believe apart from any work under the original covenant of works.
Conclusion: There are only two covenants from Adam to the end of the world, but they have undergone modifications and progressions. The first covenant broken by Adam maintains an eternal curse on everyone who has not entered into the new covenant by faith. The new covenant in every way superseded the old covenant just as the Sun takes away the light of the Moon when it rises.
Note: I should probably add scripture references to this post but I just typed it quickly as it came to mind. As each sentence can be referred to by numerous biblical references I am afraid I would have never attempted to answer as it would have taken me hours. I would not usually do so but I am relying on the reader to have some knowledge of the bible to know that each step of my argument can refer to very numerous biblical references. I have done this as the question really requires a broad brush of scripture as it is a very difficult and broad question.
Here is a list of the covenants which the Bible explicitly describes God making. It is possible that there are other covenants (such as with Adam), but the Bible does not use covenantal language to talk about them.