The first thing that comes to mind, is that you could create a SP
Designer workflow that automatically puts the "created by" information
in that "Assigned To" field  when the item  is created.

This method alone would still show the "assigned" to field when they
create an item, but whatever they select, it would change  to themselves
when they save it for the first time.

A way that you could hide the "Assigned To" field (only if necessary)
would be to enable the use of content types in that list, which would
allow you to show and hide particular fields.

Create a Custom field  (inherits from Microsoft.SharePoint.SPFieldUser)

Overwrite DefaultValue property (public override string DefaultValue)

and return web.CurrentUser value as a default  value from this property.

Add this new field in fldtypes*.xml file and use this field for ur "Person
or Group" column.

The current logged in user  id will be the default value for this field in
New item  form.