Print "ORIGINAL" first time. Then "COPY".

You will have to create a non-printable button with a custom action for printing and a pint only text field who's value will be changed by your print action.

Based on George's suggestion and a similar hurdle I was working on I have found a solution that may work for you. This is for an ORIGINAL and COPY #1 and COPY #2. The process will cycle through the three notations and return the form back to the ORIGINAL notation.

ALL CAPS INDICATE MENU SELECTIONS or EMPHASIS

Start by opening your form and edit the form FORMS>ADD OR EDIT FIELDS...Create four hidden TEXT fields and assign them the following properties in the Text Field Properties dialog (make only these adjustments anything not mentioned leave as the default setting)

FIRST HIDDEN TEXT FIELD
GENERAL TAB
NAME: text1
FORM FIELD: Hidden

APPEARANCE TAB:
Set whatever font values you want

OPTIONS TAB
DEFAULT VALUE: "ORIGINAL" (without the quotes)
Check whatever boxes you want

Click CLOSE

SECOND HIDDEN TEXT FIELD
GENERAL TAB
NAME: text2
FORM FIELD: Hidden

APPEARANCE TAB:
Set whatever font values you want

OPTIONS TAB
DEFAULT VALUE: "COPY #1" (or whatever you want and without the quotes)

Check whatever boxes you want

Click CLOSE

THIRD HIDDEN TEXT FIELD
GENERAL TAB
NAME: text3
FORM FIELD: Hidden

APPEARANCE TAB:
Set whatever font values you want

OPTIONS TAB
DEFAULT VALUE: COPY #2 (or whatever you want and without the quotes)
Check whatever boxes you want

Click CLOSE

FOURTH HIDDEN TEXT FIELD
GENERAL TAB
NAME: text4
FORM FIELD: Hidden
APPEARANCE TAB:
Set whatever font values you want

OPTIONS TAB
DEFAULT VALUE: "ORIGINAL" (without the quotes)
Check whatever boxes you want

Click CLOSE

As George suggested, create a BUTTON (but in this case it is printable) and place it wherever you need it on the form. Assign the BUTTON the following properties in the Button Properties dialog...make only these adjustments anything not mentioned leave as the default setting.

GENERAL TAB
NAME: button1
FORM FIELD: Visible
READ ONLY: checked

APPEARANCE TAB:
Set whatever font values you want

OPTIONS TAB
LAYOUT: Label only
BEHAVIOR: None
LABEL: "ORIGINAL" (without the quotes)

Click CLOSE

Stop editing the form FORMS>CLOSE FORM EDITING
.
Go to Set Document Actions ADVANCED>DOCUMENT PROCESSING>SET DOCUMENT ACTIONS...
.
Starting with the last action in the WHEN THIS HAPPENS... box and working upward:
.
Select DOCUMENT DID PRINT
Click EDIT
Type or paste in the following:
this.getField("button1").buttonSetCaption(getField("text2").valueAsString);
Click OK
.
Select DOCUMENT WILL PRINT
Click EDIT
Type in a single space: (in other words hit the space bar)
Click OK
.
Select DOCUMENT DID SAVE
Click EDIT
Type or paste in the following:
this.getField("button1").buttonSetCaption(getField("text3").valueAsString);
Click OK
.
Select DOCUMENT WILL CLOSE
Click EDIT
Type or paste in the following:
this.getField("button1").buttonSetCaption(getField("text4").valueAsString);
Click OK
.
Click OK to exit the Document Actions dialog
.
DO NOT SAVE THE FORM!!!! YET
.
We did not use the text1 hidden text field...more on that later.
.
Now, with your form open enter any required data in the visible fields but, DO NOT SAVE THE FORM!!!! YET
.
The BUTTON should be displaying as ORIGINAL
.
Print the form FILE>PRINT. The form will print with the ORIGINAL notation, but as soon as the screen returns from the PRINT dialog the BUTTON will change to COPY #1.
.
Print the form FILE>PRINT. The form will print with the COPY #1 notation and when the screen returns from the PRINT dialog the BUTTON will still display as COPY #1. On this step if you need multiple copies of the non-original then adjust the number of copies in the PRINT dialog, and you may want to change the label COPY #1 to simply COPY.
.
Save the form FILE>SAVE. DO NOT USE SAVE AS! When the screen returns from the SAVE dialog the BUTTON will change to COPY #2.
.
Print the form FILE>PRINT. The form will print with the COPY #2 notation, but as soon as the screen returns from the PRINT dialog the BUTTON will change to COPY #1.
.
Clear the form if you like FORM>CLEAR FORM and the BUTTON will change to ORIGINAL.
.
Close the form FILE>CLOSE. When prompted select DON'T SAVE!!!!
.
If you open the form back up, the BUTTON will display as ORIGINAL and you can print another set of forms with different data.
.
I know these instructions were wordy, but wanted to make this solution available to any level of Acrobat user without having to point you to some arcane Javascript document. And yes, I never did use the hidden TEXT field text1, but I need it in my own particular solution and did not want to alter a working version of the solution.

I do not know down to what version of Acrobat this works, I use 8, 9 and X.

Instead of continually having to perform these steps again and again, you can use a Quickeys action or some such other program to repeat keyboard steps.