What is the minimum weight aircraft in which someone can build logged flight time?
There is no weight limit, requirement, nor restriction for logging time in a logbook. However, there is a requirement to only fly aircraft that are properly certificated, registered, and airworthy regardless of if you log the time or not. Part 103.7 allows ultralights to fly without certification or registration. If you are flying an aircraft not meeting certification, registration, inspection, and airworthiness standards, you are flying illegally unless it meets the weight, horsepower and other criteria of Part 103.1.


  §103.7 Certification and registration.
  (a) Notwithstanding any other section pertaining to certification of aircraft or their parts or equipment, ultralight vehicles and their component parts and equipment are not required to meet the airworthiness certification standards specified for aircraft or to have certificates of airworthiness.
  
  (b) Notwithstanding any other section pertaining to airman certification, operators of ultralight vehicles are not required to meet any aeronautical knowledge, age, or experience requirements to operate those vehicles or to have airman or medical certificates.
  
  (c) Notwithstanding any other section pertaining to registration and marking of aircraft, ultralight vehicles are not required to be registered or to bear markings of any type.


Combining this with Part 61.51(j) would prohibit the logging of ultralight experience gained today for the purpose of initial certification or currency.


  (j) Aircraft requirements for logging flight time.
  For a person to log flight time, the time must be acquired in an
  aircraft that is identified as an aircraft under §61.5(b), and is—
  
  (1) An aircraft of U.S. registry with either a standard or special airworthiness certificate;
  
  (2) An aircraft of foreign registry with an airworthiness certificate that is approved by the aviation authority of a foreign country that is a Member State to the Convention on International Civil Aviation Organization;
  
  (3) A military aircraft under the direct operational control of the U.S. Armed Forces; or
  
  (4) A public aircraft under the direct operational control of a Federal, State, county, or municipal law enforcement agency, if the flight time was acquired by the pilot while engaged on an official law enforcement flight for a Federal, State, County, or Municipal law enforcement agency.


Furthermore, Part 61.52 does not allow for the logging of ultralight experience gained today for the purpose of initial certification or currency.


  §61.52 Use of aeronautical experience obtained in ultralight vehicles.
  (a) Before January 31, 2012, a person may use aeronautical experience obtained in an ultralight vehicle to meet the requirements for the following certificates and ratings issued under this part:
  
  (1) A sport pilot certificate.
  
  (2) A flight instructor certificate with a sport pilot rating;
  
  (3) A private pilot certificate with a weight-shift-control or powered parachute category rating.
  
  (b) Before January 31, 2012, a person may use aeronautical experience obtained in an ultralight vehicle to meet the provisions of §61.69.
  
  (c) A person using aeronautical experience obtained in an ultralight vehicle to meet the requirements for a certificate or rating specified in paragraph (a) of this section or the requirements of paragraph (b) of this section must—
  
  (1) Have been a registered ultralight pilot with an FAA-recognized ultralight organization when that aeronautical experience was obtained;
  
  (2) Document and log that aeronautical experience in accordance with the provisions for logging aeronautical experience specified by an FAA-recognized ultralight organization and in accordance with provisions for logging pilot time in aircraft as specified in §61.51;
  
  (3) Obtain the aeronautical experience in a category and class of vehicle corresponding to the rating or privileges sought; and
  
  (4) Provide the FAA with a certified copy of his or her ultralight pilot records from an FAA-recognized ultralight organization, that—
  
  (i) Document that he or she is a registered ultralight pilot with that FAA-recognized ultralight organization; and
  
  (ii) Indicate that he or she is recognized to operate the category and class of aircraft for which sport pilot privileges are sought.


Weight is not a factor in logging time. And, there is no prohibition against putting things in your logbook that can not be counted for the purposes of initial certification or currency. Although, care should be taken with this. Your logbook is your property. However, your certification gives the FAA the right to inspect your logbook however it chooses. I have co-pilot time logged in a turbine-powered helicopter that I can never count towards anything.

There is also no prohibition against getting an ultralight registered and certificated as airworthy. It might be a lengthy and expensive (and probably improbable) endeavor. But, as soon as you do, it becomes an ultralight no longer according to Part 103.1


  §103.1 Applicability.
  This part prescribes rules governing the operation of ultralight vehicles in the United States. For the purposes of this part, an ultralight vehicle is a vehicle that:
  
  (a) Is used or intended to be used for manned operation in the air by a single occupant;
  
  (b) Is used or intended to be used for recreation or sport purposes only;
  
  (c) Does not have any U.S. or foreign airworthiness certificate; and
  
  (d) If unpowered, weighs less than 155 pounds; or
  
  (e) If powered:
  
  (1) Weighs less than 254 pounds empty weight, excluding floats and safety devices which are intended for deployment in a potentially catastrophic situation;
  
  (2) Has a fuel capacity not exceeding 5 U.S. gallons;
  
  (3) Is not capable of more than 55 knots calibrated airspeed at full power in level flight; and
  
  (4) Has a power-off stall speed which does not exceed 24 knots calibrated airspeed.