What are good strategies or principles to prolonging the first stage of a game? Is there a particular strategy or principle you can use to drag out the first stage of a game, namely, to preserve all your pieces as long as possible.

One way I can imagine this being useful, if early on you exchange your bishop for an opponent's knight, having a crowded board would thus become in your best interest, as knights are generally more valuable on crowded boards. (Or so I am told)

So once again, the goal is to lose as few pieces as possible for the longest stretch of time. 

Some questions were posed to me for clarification:

Why do you want to avoid exchanges? 

To see if it is possible to avoid exchanges without hurting your positioning, or by only hurting it in an insignificantly marginal manner.

What is your goal? 

To prolong the game. Perhaps if it were a game of speed-chess, and you wished to run down the clock to play for a draw; this could be a good strategy. Perhaps if you knew your opponent to be impatient, this could be effective against him.

Do you aim for a mating attack? 

No.

Do you need a middlegame position that gives lots of maneuvering and has a high chance of dodging the endgame ( maybe you do not like to play endgame )?

Great Question; This would be ancillary.
Is there a particular strategy or principle you can use to drag out
  the first stage of a game?
  
  One way I can imagine this being useful, if early on you exchange your
  bishop for an opponent's knight, having a crowded board would thus
  become in your best interest, as knights are generally more valuable
  on crowded boards. (Or so I am told)


This entire question is unclear, but I will give it a try to answer. Let us start one by one:


  Is there a particular strategy or principle you can use to drag out
  the first stage of a game?


Now why would you want to do that?


  One way I can imagine this being useful, if early on you exchange your
  bishop for an opponent's knight, having a crowded board would thus
  become in your best interest, as knights are generally more valuable
  on crowded boards. (Or so I am told)


Aha! So you wish to keep the board crowded in order to exploit the "advantage" knight has over a bishop!

Well, it is time to answer:

You can not keep the board crowded unless both you and your opponent "agree" to do so.

He can always try to exchange pieces and reduce the tension if he chooses, and the best thing you can do is to try and stop him by making the exchanges unfavorable for him.

Unfortunately, the opening theory has advanced a lot, and there are ways for both sides to obtain a good game, so the above plan will be very hard, if not impossible, to execute. 

Another thing: simply exchanging the bishop for a knight, and trying to keep the pieces on board will not give you an advantage. It will all boil down to the placement of your pieces. By the way, exchanging the bishop for a knight early leaves your opponent with the bishop pair, which may give him the advantage.

I would strongly suggest you to find a resource ( book/online tutorial ) for playing the opening.

EDIT IN RESPONSE TO THE OP's EDIT:


  Why do you want to avoid exchanges?
  
  To see if it is possible to avoid exchanges without hurting your
  positioning, or by only hurting it in an insignificantly marginal
  manner.


This is not possible, I am afraid. You will have to exchange at some point and here is why:

When amateur wants to exchange pieces, he uses his own to "chase" the opposing one until the exchange has been made.

Master, on the other hand, puts the piece he wants to exchange on the strong squares, thus forcing the exchange. That way the piece becomes annoying and strong and opponent is forced to exchange it or else he will stand worse.

In a strong game, versus strong opponent you will not be able to dodge the exchange.


  What is your goal?
  
  To prolong the game. Perhaps if it were a game of speed-chess, and you
  wished to run down the clock to play for a draw; this could be a good
  strategy. Perhaps if you knew your opponent to be impatient, this
  could be effective against him.


This is very much possible, but you will need to play typical middlegame positions in order to achieve this. Furthermore, you will need to be the one who dictates the game. This is very important requirement. 

You can play positions with an isolated pawn. My answer provides comprehensive coverage of the plans for both sides. Still, let me remind you-side with the isolated pawn exchanges pieces only when it can get a definite, visible-with-the-naked-eye benefit. As long as the board is packed with pieces the advantage is on its side. Side without isolated pawn strives to exchange minor pieces and enter endgame. Again, in the endgame you can prolong the game as much as you want as you will have the advantage. Usually you will be the only one who can play for a win and your opponent usually will have no counterplay but will be forced to passively defend.

Another typical position that has a lot of maneuvering and is slow is Carlsbad pawn structure and it arises from Queen's Gambit Declined after White exchanges on d5 with the move cxd5. White again has lots of plans here, but you would be interested in minority attack. Minority attack creates a weak pawn for Black on c6 and allows White to perform long maneuvers in order to take it ( just don't underestimate Black's mating attack on the kingside! ). This would be ideal choice for you.

You could try French defense with both colors. With White you can try Advance variation, and with Black Winaver sounds good. Again, both positions require long maneuvers.

END OF EDIT:

Best regards.