This is a common practice. Since the harvester of hay has your father's consent, there is no adverse possession. It may be good to get a simple one line - one page agreement signed by the harvester stating that in exchange for the mowing, he is entitle to keep the hay. There is so little profit to be made, I doubt that you will find anyone who pays to harvest. That simple agreement should cover the bases.
Actively practicing law in Texas. Inactive licenses in Arizona and Georgia. All answers are general in nature and no attorney/client relationship exists in this forum.
There are really two legal issues here, one a land use question, the other a contracts issue. With regards to the land use question, based on this information, no, the laborer has no property rights in your father's property. To acquire rights in land that is not yours by working on it or occupying it, a person must use the property adversely to the owner in an open, notorious, and continuous way for a period prescribed by state statute, which, in Pennsylvania, is 21 years. 42 P.a.C.S. Section 5530, see cite below.
The laborer is not using the property adversely but rather with your father's permission. Permission to use property usually creates a revocable license, which means that your father may revoke the license and eject the licensee (the laborer) at any time.
HOWEVER, you also asked whether the laborer has any legal right to demand "compensation," which could include monetary compensation. To determine the answer to the compensation question, you would have to examine a fairly broad range of contract principles because many contracts do not have to be in writing to be enforceable. To create a contract, all that is required is: an offer, acceptance of that offer, consideration, and an intention by both parties to the contract to be legally bound.
Those are probably not all terms a layperson is familiar with. Even after three years of law school and many years of practice, not many lawyers, including this one, can fully explain these concepts without allowing for some ambiguity. In the context of your question, explaining these issues in a way that will give you some guidance is hard because the precise terms of this unwritten agreement between your father and the laborer are not spelled out.
For instance, who benefits from the hay harvest? I am presuming it is your father's hay. Who sells the hay? Your father undoubtedly benefits from the soil treatment, tree removal, and landscaping. The laborer benefits from the storage space, but is the use of the storage space only necessary because these folks are performing labor on the property? It seems like you are saying the laborer gets nothing out of this deal that is of value. That is important because that is what "consideration" means.
"Consideration" means something of value given by a party to a contract that is the inducement for the other party to enter the contract. If your father has offered nothing of value to the laborer except the opportunity to work in the sunshine, there is no consideration and therefore no contract.
If there is a portion of this relationship that results in some benefit to the laborer, then the laborer would likely have a legal right to enforce the terms of the agreement to gain the benefit. However, to sue upon a contract, either express or implied, a breach of the contract must occur. If your father has already performed his side of the contract, he has not breached it and likely would not breach this implied contract if he said, "No more work on my land." Say, however, that I assume that the agreement here is actually the following: Your father has agreed to allow the laborer to harvest the hay and sell it. Your father kicks the laborer off the property mid-harvest. The laborer could sue under a breach of contract theory. That hypothetical is pretty far off from your question, I think, but we are being comprehensive here.
Also, if the arrangement were such that your father would allow the laborer to sell the hay, the laborer could claim that he or she has a lease allowing the work and the relationship to continue. A lease would be subject to the requirement that it be in writing, but in Pennsylvania it appears that only leases of more than three years are required to be in writing. See the link below.
In summary, if the laborer truly is working for nothing, your father is likely free to eject him without liability. If the laborer was reasonably expecting some benefit from the arrangement, your father may not be free to eject.
I agree with the other two opinions on adverse possession. You have not described adverse possession. For more on that topic see: http://tupitzalaw.com/adverse-possession/
There is a second topic: ownership of the crop. Generally hay is just grass and is not like corn or soybeans. However, if the person has recently fertilized or spread lime, you may need to give them time to amortize the expense.
Consider a written lease. As Robert Frost said "Good Fences make Good Neighbors." I have a written lease with the man who farms some of my fields so we both know what will happen if I change my mind about farming.
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