Lukas. 15:29 KJV
And he answering said to his father, Lo, these many years do I serve thee, neither transgressed I at any time thy commandment: and yet thou never gavest me a kid, that I might make merry with my friends:
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This is a good question because we often concentrate on the father's gracious attitude toward his profligate son and pay little attention to the elder son. The OP author mentions in his comment that he wonders if the parable is related to birthright inheritance. I think the answer is yes.
The son who stayed on the farm is the elder son. He is in the position of Cain and of Esau - both being elder sons who experienced similar feelings of unfair treatment, either by God or by their human father. The elder son, like Cain and Esau, felt that the person in the father's position had given his blessing to the younger son with no good reason. Both Cain and Esau brought offerings and hoped to be blessed but God and Isaac accepted the animals offered by Abel and Jacob. The fatted calf thus echoes Abel's and Jacob's offerings. The calf and the ring also harken back to the story of Judah, who gave his ring and staff to the disguised Tamar together with a promise to send her a young goat.
Each of these stories is tied to the theme of lineage and/or birthright blessing. So my answer is yes, the story of the prodigal is indeed related to the idea of birthright blessing. In the history of the people of Israel, the chosen lineage rather consistently favored younger candidates for God's blessing, both in the time of the patriarchs and the time of kings (Saul and David, Adonijah and Solomon etc.) In the context of the time of Luke's writing, there is also a message to Jesus' pious Jewish followers (e.g. Acts 15:5), who had tried to remain faithful to God's law throughout their lives. Its lesson is that Jewish Christians, in the position of elder brother to gentile converts, should welcome former sinners and even idol-worshiping pagans home as the Father did in the parable.
Conclusion: the parable is indeed related to the issue of God's blessing the younger brother, a consistent theme in OT stories. The parable's main point is the grace of the father (representing God) toward his sinful son (representing sinners generally), but its secondary point is that God often blesses the younger and pays the same "wage" to those who come late to work as those who started early. (Matthew 20: 1–16) Pious elders should share the Father's joy at the homecoming of less worthy youngers. The "last" may have become the "first," but they have not lost Father's love: "you are here with me always; everything I have is yours." (Lk. 15:31)
The question: "What does 'neither transgressed I at any time' mean in the parable of the prodigal son?" appears to be a question about language. (Perhaps the author may wish to embellish the question.) As it stands, I will claim that King James English can sound a little like Yoda-speak sometimes.
Updating the language for 'Neither transgressed I' would yield something like 'I never transgressed' where Webster's Dictionary gives the two definitions for 'transgress':
The elder son is complaining. He is stating that he never went against his father's wishes, and despite his obedience was never given the means (see note below) to throw a party with his friends, as was being done for the rebellious son.
Note: a kid or calf could be roasted on an open spit, and would make for quite a feast!