My inspiration for this study comes from Ps. 105:8-15, especially v. 15. The context of this passage is the covenant that God made with Abraham, Isaac, and Jacob (Abramic Covenant; cf. Gen. 12-13, 15, 17, 22). So the antecedent of "mine anointed" and "my prophets" in v. 15 is these three patriarchs. I would like to now offer scriptural support that these men were not only prophets, but also priests and kings, making them all, perhaps, types of the Lord Jesus Christ to varying degrees, the ultimate prophet-priest-king (Deut. 18:18; Ps. 110:4; Rev. 1:5-6).
1. Prophets. (a) God calls Abraham a prophet in Gen. 20:7, even after he lies to Abimelech about Sarah. I think that an example of Abraham prophesying is found in Gen. 25:22-23, where Rebekah goes to enquire of the LORD. Abraham was still alive at this time (he fathers Isaac at 100, Jacob and Esau are born when he's 160, and he dies at 175; cf. Gen. 25:7, 20, 26), and was there really a need for Rebekah to go to anyone else, since her father-in-law was a prophet? Melchizedek was a king-priest (Gen. 14:18), but not said to be a prophet. (b) Isaac prophesies in Gen. 27:27-29, 37-40 concerning the futures of Jacob and Esau (Heb. 11:20). (c) Jacob prophesies concerning Joseph's sons, Manasseh and Ephraim (Gen. 48:14-20; Heb. 11:21), and then concerning his own twelve sons (Gen. 49).
2. Priests. Even though Melchizedek is called "the priest of the most high God" (Gen. 14:18), I would assert that priesthood was not limited to him at this point. Each of the patriarchs serve as priests of their households, with each of them building altars and offering sacrifices at various points in their lives (cf. Gen. 12:8, 13:4, 15:9-11, 22:1-19 (Abraham); 26:15 (Isaac); Gen. 33:20, 35:7 (Jacob)). Furthermore, the Levitical priesthood springs from the patriarchs (Heb. 7:9-11), rather than Melchizedek, whose priesthoood is distinctive from the Levitical, according to both testaments (Ps. 110:4; Heb. 5, 7).
3. Kings. This office may not stand out like the others do, but there is scripture to support it. When God called Abram in Gen. 11-12, he effectually made him the king of the world, since the fate of "all families of the earth" depended on their treatment of Abram, making him, effectually, their head (Gen. 12:1-3). Paul refers to Abraham as the "heir of the world" in Rom. 4:13, and Moses tells the Israelites in Deut. 4:40 that God gave them "the earth...for ever." Hence the reference (perhaps indirect) to part of the underworld as "Abraham's bosom" (Luke 16:23), since the earth was given to Abraham. So Abraham was "a king without a throne," as we say. Finally, in Ps. 105:14-15, we see the royalty of Abraham, Isaac, and Jacob mentioned, since they are referred to as "mine anointed" in contrast to the "kings" in v. 14 (remember King Saul, "the LORD's anointed"; cf. 1 Sam. 24:5-6). Granted, anointing was also connected with the prophetic (1 Kings 19:16) and priestly (Lev. 6:20, Num. 35:25) offices, but in context "mine anointed" obviously speaks of their royalty and, perhaps, secondarily, of their prophetic office.
Abraham, Isaac, and Jacob were so much more than wandering shepherds, weren't they?!