How high can I fly a kite above my farm?
It depends how big the balloon or kite is. Balloons and kites are governed by 14 CFR 101, but specific regulations only apply above a certain size. 101.1 says:


  a) This part prescribes rules governing the operation in the United
  States, of the following:
  
  (1) Except as provided for in §101.7, any balloon that is moored to
  the surface of the earth or an object thereon and that has a diameter
  of more than 6 feet or a gas capacity of more than 115 cubic feet.
  
  (2) Except as provided for in §101.7, any kite that weighs more than 5
  pounds and is intended to be flown at the end of a rope or cable.


However, note the reference to 101.7, which says (in part):


  No person may operate any moored balloon, kite, amateur rocket, or
  unmanned free balloon in a manner that creates a hazard to other
  persons, or their property.


That means, if your balloon or kite is too small to be governed by part 101 then there are no specific restrictions on how you operate it, but you still can't fly it in a way that puts other people in danger.

If part 101 does apply to you, then you have to follow the rules in 101.13 and 101.15:


  §101.13   Operating limitations. (a) Except as provided in paragraph (b) of this section, no person may operate a moored balloon
  or kite—
  
  (1) Less than 500 feet from the base of any cloud;
  
  (2) More than 500 feet above the surface of the earth;
  
  (3) From an area where the ground visibility is less than three miles;
  or
  
  (4) Within five miles of the boundary of any airport.
  
  (b) Paragraph (a) of this section does not apply to the operation of a
  balloon or kite below the top of any structure and within 250 feet of
  it, if that shielded operation does not obscure any lighting on the
  structure.
  
  §101.15   Notice requirements. No person may operate an unshielded moored balloon or kite more than 150 feet above the surface of the
  earth unless, at least 24 hours before beginning the operation, he
  gives the following information to the FAA ATC facility that is
  nearest to the place of intended operation: [contact details, time, altitude etc.]


There are more requirements in 101.17 and 101.19.