user11144: We assume standard (P1BN3R5Q8) value (personally I prefer 1,3,3,4.5,9 but so there).
To get a drawn position with maximum material difference, not caring about that it's illegal, we just have to plaster the field with bQs and shield the wK in the corner with a bR diagonally, plus bB or bN at the sides. This position is an immediate stalemate, no matter who has the move. (Value: 3+3+5+(64-5)*8=483)
"Improve" this position, still maximum material difference, such that the position can be played forever with help play (ignoring the dead, repetition, and 50 move rule which will cut this "play" short in this order!). This includes the possibility that Black has enough at some time and stalemates White (or self after capture) after all. Stalemate just may not be forced. Feel free to choose who has the move, but choose wisely (insert fiendish laughter). (Feel also free to give an additional position where no side can force a stalemate on self or other).
For FEN convenience, you may just omit the bQs from the diagram.

user8860: If I count right we actually do a bit better than Hauke Reddmann's
construction by giving White a pawn and the initiative:
[FEN "kqqqqqqq/Pqqqqqqq/qqqqqqqq/qqqqqqqq/qqqqqqqq/qqqqqqqq/qqqqqqrb/qqqqqqbK w KQkq - 0 1"]

    1.axb8=R+! Ka7! 2.Rxb7+! Ka8! 3.Ra7+

White has an immediate forced draw by 1 axb8=Q+!,
but play may instead continue 1 axb8=R+! Ka7! 2 Rxb7+! Ka8! 3 Ra7(b8)+ etc.
as shown, with draw by perpetual check;
none of Black's (or for that matter White's) alternatives actually win,
but the "!" are for the unique moves that allow the game to continue forever.

user11144: I don't think this position can be improved, but feel free to show me the contrary (note that the double check is retroillegal too, but who cares in an illegal position!):
[FEN "Krkqqqqq/q1pqqqqq/pprqqqqq/qqqqqqqq/qqqqqqqq/qqqqqqqq/qqqqqqqq/qqqqqqqq w KQkq - 0 1"]

Game may continue 1.Kxa7 Rb8+ 2.Ka8 Rb7+ forever, until Black selfstalemates with Ra8+ or Ra7+ at any point.

11144: <p>We assume standard (P1BN3R5Q8) value (personally I prefer 1,3,3,4.5,9 but so there).</p>
<p>To get a drawn position with maximum material difference, not caring about that it's illegal, we just have to plaster the field with bQs and shield the wK in the corner with a bR diagonally, plus bB or bN at the sides. This position is an immediate stalemate, no matter who has the move. (Value: 3+3+5+(64-5)*8=483)</p>
<p>&quot;Improve&quot; this position, still maximum material difference, such that the position <em>can</em> be played forever with help play (ignoring the dead, repetition, and 50 move rule which will cut this &quot;play&quot; short in this order!). This includes the possibility that Black has enough at some time and stalemates White (or self after capture) after all. Stalemate just may not be forced. Feel free to choose who has the move, but choose wisely (insert fiendish laughter). (Feel also free to give an additional position where no side can force a stalemate on self or other).</p>
<p>For FEN convenience, you may just omit the bQs from the diagram.</p>
 8860: <p>If I count right we actually do a bit better than <strong>Hauke Reddmann</strong>'s
construction by giving White a pawn and the initiative:</p>
<pre><code>[FEN &quot;kqqqqqqq/Pqqqqqqq/qqqqqqqq/qqqqqqqq/qqqqqqqq/qqqqqqqq/qqqqqqrb/qqqqqqbK w KQkq - 0 1&quot;]

    1.axb8=R+! Ka7! 2.Rxb7+! Ka8! 3.Ra7+
</code></pre>
<p>White has an immediate forced draw by 1 axb8=Q+!,
but play may instead continue 1 axb8=R+! Ka7! 2 Rxb7+! Ka8! 3 Ra7(b8)+ etc.
as shown, with draw by perpetual check;
none of Black's (or for that matter White's) alternatives actually win,
but the &quot;!&quot; are for the unique moves that allow the game to continue forever.</p>
 11144: <p>I don't think this position can be improved, but feel free to show me the contrary (note that the double check is retroillegal too, but who cares in an illegal position!):</p>
<pre><code>[FEN &quot;Krkqqqqq/q1pqqqqq/pprqqqqq/qqqqqqqq/qqqqqqqq/qqqqqqqq/qqqqqqqq/qqqqqqqq w KQkq - 0 1&quot;]
</code></pre>
<p>Game may continue 1.Kxa7 Rb8+ 2.Ka8 Rb7+ forever, until Black selfstalemates with Ra8+ or Ra7+ at any point.</p>
