It will depend upon your state's laws and fees if you will have to pay to obtain a learner's permit but in most states there is some sort of driver's license or application involved with starting the licensing process (a graduated driver's licensing process in most states) and getting your learner's permit.
For example the Virginia Department of Motor Vehicles notes that currently the normal driver's license fee is $4.00 per year. A learner's permit fee is $3.00 plus the yearly cost of a driver's license so a total of $7.00.
In Washington State there is a fee of $20 that you must pay when applying for an instruction permit.
In Pennsylvania for an initial permit and four-year license cost $33.00.
In California the application fee of $28.00 is required when you apply for a provisional permit. This fee entitles you to three exams of any type within the 12-month period and pays for both the instruction permit and the driver license. If all requirements are not met within the 12-month period, the application becomes void and all steps must be completed again.
In New York State there is one application form for both your first learner permit and your first driver license. You must pay the application fees when you visit a DMV office to complete the application. The fee amount differs depending upon your age. If you are 16 or 16 1/2 it is $75 and then the fee amount goes down the older you get so if you apply at age 21 the cost is between $52.50 and $55. For a complete list of the fee schedule of NYS you would need to contact the NYS DMV.
The Florida Department of Highway Safety and Motor Vehicles (HSMV) has on its site that it is $27 to get your first FL driver’s license. Taking the written, hearing and vision tests are all fee however if you fail the written test and must retake it, it will cost $5.00. And later on when you are progressing through the licensing system if you must retake the skills/driving examination test there is a $10 fee.
Contact your state's Department of Motor Vehicles (or like state agency) to find out if there is a fee to obtain a learner's permit and if so how much it will cost.