People do it all the time. No application then no paperwork, no need to worry about benefits, taxes, SS, etc. Hire for a job, pay in cash. No prob. Not necessarily legal but still done. Usually short term, temp jobs. Child care, yardwork, housekeeping, field hands, these and more are often done via 'under the table' or 'street corner' employment.
In my own opinion, an employer can hire an individual without him or her filling in an application because he is familiar with both your educational background your work experience. It would be advisable to ask for it just in case he has forgotten to ask you fill it to avoid conflict in the future.
I know in a handful of states I've worked in (VA, MD, VT, DC), it is illegal not to have an application or resume on file for every employee working for you. It should be the first piece of an employee's personnel file.