Since your in a MDE file you can not modify anything with code, including Modules, Reports, and Forms. So now you're left with Tables, Queries, and Macros.
Let's see...you want to open a DB only five times only.
Divide and Conquer:
(1)Create a table to store the number of times the DB has been opened.
Table Name: tblApplication
Fields: RecID Autonumber
TimesOpen Number-Integer
Create a record. Remember the RecID number and set the TimesOpen to zero. Let's say its equal to 1.
(2)Create an update query that will update TimesOpen value.
The SQL would look like the following:
UPDATE tblApplication SET tblApplication.TimesOpen = [TimesOpen]+1
WHERE (((tblApplication.RecID)=1));
(3)Create a Macro that will:
(a) Set your warnings off
(b) Run the update query that you created
(c) Set your warnings back on.
(d) CRITERIA: DLookUp("[TimesOpen]","[tblApplication]","[RecID]=1"

>5; ACTION: Msgbox to the user that the application has expired.
(e) CRITERIA: DLookUp("[TimesOpen]","[tblApplication]","[RecID]=1"

>5; ACTION: Quit (Shuts down Access)
(4)Run the macro from your Autoexec macro. If you don't have an Autoexec macro...create a macro, and name it Autoexec.
Hope this helps