Show initially 4 fields for Text field with unlimited value
Check Multivalue Settings Module


  Warning: this module is still very buggy. Actually, it only works in
  very few use cases. I suggest to only attempt using it with core
  fields (not fields from contrib modules). Even this may not work. I'm
  working on fixing things, but don't have much time. Help is welcome.


Functionality:

Number of values allowed
Number of values required
Number of empty widgets to display initially (Drupal usually displays all fields for non-unlimited fields. This allows you to only display a few, followed by the 'Add More' button, which will disappear after the maximum number of fields has been filled.)
Number of empty widgets to display on edit forms after the field contains some values (Drupal usually displays one extra)


Though the module is buggy but can be a good starting point for building a custom solution or patching the same module.