Basically, as soon as the computer finishes you (and it says 'Finish him)
hit the start button for the other player (ie. if you are player 1, hit F2).
This does not cost a credit and player 1 does not have to hit start so does
not have to use a credit.  The winner of the fight (just screw around to
get it over quickly) continues to fight the computer and has not
lost a credit.

Of couse your score and 'wins in a row' is reset - but its reset if you
use a credit anyways

The short of it:
1) Load game
2) Press F10 for setup, hold SHIFT while typing D, I, and then P (DIP).
3) Turning SWITCH 5 to ON will give you FREE PLAY (unlimited continues)
4) The change will take effect instantly, no need to save changes
   (unless you want to keep it in the config file) or reload.
   (However, see #4 below to find out about what kind of changes
   require reloading)
 
Here are the intense details:
 
1) I have tested this on MK v1.0 for the IBM PC (both with and without
   the cheat patch by Stimpson J. Cat) as well as "v1.1", the 1/4/94
   version updated by the MKFIX.EXE from Acclaim.  So it basically
   seems to work on any "release" version, but I did not test it on
   any of the beta releases.  However, since the cheat involves the
   "F10 Setup Screen" I highly doubt it works on the beta version
   which had a separate SETUP.EXE.  They were tested with a
   Soundblaster v1.0, 486/33, 8 megs RAM.  (This may seem irrelevant
   but one of the below Switches does affect sound card output)
 
2) Load the game.  Go to the setup screen by hitting F10.  At the
   setup screen, hold SHIFT, and press the letters "D", "I", and then
   "P" in that order.  It doesn't matter if you hold left or right
   shift, as some people were discussing.  You don't hold D-I-P
   simultaneously or anything, just hold SHIFT the whole time then
   basically type "DIP".
 
3) A screen will pop up with this list:
 
        SWITCH 0: ON
        SWITCH 1: ON
        SWITCH 2: ON
        SWITCH 3: OFF
        .(etc)
        .(etc)
        SWITCH 8: OFF
 
   The default is with those first 3 switches on, and the rest off.
 
4) Now about these switches.  For one thing, SOME of the switches are
   a real pain in the ass to test through playing the game because
   they do NOT take effect until you reload the game after "EXIT &
   SAVE" from the SETUP screen.  Basically, things like FREE PLAY do
   not require reloading the game since they do not require loading a
   set of graphics or sound, but things like turning
   fatalities/blood/sound on or off DO require reloading the game
   since they require loading a different set of graphics or sound.
   Here are a FEW things I have found out from testing the DIP
   switches so far.
 
   NOTE: All of these tests were done using Sub Zero, and knowing
   those bastards at Midway they might have different effects
   depending on who you use.  Also, aside from all switches ON or all
   OFF, I did not test like "Switches 1,4,7 ON, all the rest off" or
   anything like that.  Again, certain switches being ON/OFF in
   combination might very likely produce secret effects.  Finally,
   with each individual switch, just because I said it only affects
   one thing (I.E. SWITCH 0=Fatality), doesn't mean it might not
   affect something else.  I only mentioned the first major thing I
   noticed in certain cases.
 
   ALL SWITCHES SET TO OFF:
        No blood, no fatalities, the text of MIDWAY PRESENTS is yellow
        rather than white, etc mentioned individually below
 
   ALL SWITCHES SET TO ON:
        Blood, fatalities, FREE PLAY (YES!), comic book ad shown, etc.
        mentioned individually below
        
        (Each of the below was tested with SWITCH x: ON, all the rest
         OFF, except as noted.)
 
   SWITCH 0: (does not affect blood during match) (default ON)
        ON=Fatalities ON
        OFF=Fatalities OFF
 
   SWITCH 1: (does not affect fatalities) (default ON)
        ON=Blood ON (during match)
        OFF=Blood OFF
 
   SWITCH 2: (default ON)
        ON=Sound ON during title sequence (does not affect sound in game)
        OFF=Sound OFF during title sequence
 
   SWITCH 3: (default OFF)
        ON=Music ON, Sound Effects OFF (speech off,etc. I have a
                                        strong feeling this is equal
                                        to "force Adlib mode")
        OFF=Music ON, Sound Effects ON
 
   SWITCH 4: (default OFF)
        ON=CPU does Fatality EVERYTIME he/she beats you (Switch 0 must
           be on)
        OFF=Normal/Random
 
   SWITCH 5: (default OFF)
        ON=Free Play ON
        OFF=Free Play OFF (Credits=4)
 
   SWITCH 6: (default OFF)
        UNKNOWN
 
   SWITCH 7: (default OFF)
        UNKNOWN
 
   SWITCH 8: (default OFF)
        CPU recovers slowly from an attack
 

Hex editing will let you play unlimited/continuous games:

Hex edit MK.EXE
Search for A7 0C 03 FF FA 04
Change to: A7 0C 03 FF FA 0A