user-1: Consider the following chess variant:

Black can, once per game, make two moves in a row (the second move may not capture the King)
To compensate for this advantage, Black gets a material handicap
Otherwise, the standard rules of chess apply

The question is, approximately what material handicap would be needed for the game to become roughly balanced? It is clear that making two consecutive moves, even only once per game, is a tremendous advantage. My guess is that at least a Queen handicap is necessary, since it is trivially easy to capture White's Queen with two consecutive moves once the pieces are out.
Has this chess variant been studied before?

user4362: My feeling is that a queen would be too much. 
My reasoning is that without a queen it is very unlikely that the double move will ever lead to a winning attack. So at one point the double move will have to be used to "win" the queen, but if white takes care to always protect the queen this will probably lose some material.
Something materially equivalent like both rooks strikes me as better, because it doesn't impede blacks dynamical chances in the same way. Possibly rook+knight or rook+bishop would be even more balanced.

user7596: The question can never be answered in a general fashion. Every move has its own perks and advantages. The major variables are these:
Is the same piece allowed to move twice or can black move two pieces once?
Are captures allowed? With just one move of the king's pawn already made, it is possible for any party to capture the opponent's queen within two moves. (Or put them in a position where it cannot escape capture in the next few moves). Here the price for two moves is AT LEAST a queen (since the two moves party would not only have captured the queen, but also invaded inside the enemy ranks).
Starting from midgame, two moves can easily put the opponent in a mate-in-x position where they cannot avoid mate. For example, if one has several mating combinations available, there can be no possible compensation to that!
All in all, I think the option of two moves can/should be made available only in the opening and the price is the queen. It is disastrous and potentially game-losing to allow the opponent double moves in the middlegame or endgame and there can be no compensation to that.

-1: <p>Consider the following <strong>chess variant:</strong></p>

<ul>
<li>Black can, <em>once per game</em>, make two moves in a row (the second move may not capture the King)</li>
<li>To compensate for this advantage, Black gets a material handicap</li>
<li>Otherwise, the standard rules of chess apply</li>
</ul>

<p>The question is, <strong>approximately what material handicap would be needed for the game to become roughly balanced?</strong> It is clear that making two consecutive moves, even only once per game, is a tremendous advantage. My guess is that at least a Queen handicap is necessary, since it is trivially easy to capture White's Queen with two consecutive moves once the pieces are out.</p>

<p>Has this chess variant been studied before?</p>
 4362: <p>My feeling is that a queen would be too much. </p>

<p>My reasoning is that without a queen it is very unlikely that the double move will ever lead to a winning attack. So at one point the double move will have to be used to "win" the queen, but if white takes care to always protect the queen this will probably lose some material.</p>

<p>Something materially equivalent like both rooks strikes me as better, because it doesn't impede blacks dynamical chances in the same way. Possibly rook+knight or rook+bishop would be even more balanced.</p>
 7596: <p>The question can never be answered in a general fashion. Every move has its own perks and advantages. The major variables are these:</p>

<p>Is the same piece allowed to move twice or can black move two pieces once?</p>

<p>Are captures allowed? With just one move of the king's pawn already made, it is possible for any party to capture the opponent's queen within two moves. (Or put them in a position where it cannot escape capture in the next few moves). Here the price for two moves is AT LEAST a queen (since the two moves party would not only have captured the queen, but also invaded inside the enemy ranks).</p>

<p>Starting from midgame, two moves can easily put the opponent in a mate-in-x position where they cannot avoid mate. For example, if one has several mating combinations available, there can be no possible compensation to that!</p>

<p>All in all, I think the option of two moves can/should be made available only in the opening and the price is the queen. It is disastrous and potentially game-losing to allow the opponent double moves in the middlegame or endgame and there can be no compensation to that.</p>
