Busy Brain Beaver reboot
22222222... (65536 twos) = 2↑↑↑4
Before I begin, this is not the largest number expressible using 500 bytes of New Brainfuck.  It's simply where I gave up trying to wrap my head around how big the numbers were getting.  This answer only uses 93 bytes and is therefore not the largest number, but it's proof of concept that mind-bogglingly large numbers are possible.
+++[-&gt;++[-&gt;+&gt;+&lt;&lt;]&gt;&gt;[-&lt;&lt;+&gt;&gt;]+&lt;[-&gt;[-&lt;&lt;[-&gt;&gt;&gt;+&gt;+&lt;&lt;&lt;&lt;]&gt;&gt;&gt;&gt;[-&lt;&lt;&lt;&lt;+&gt;&gt;&gt;&gt;]&lt;&lt;]&gt;[-&lt;+&gt;]&lt;&lt;]&gt;[-&lt;+&gt;]&lt;&lt;&lt;]&gt;&gt;.

This is a modification of the exponentiation algorithm in Brainfuck.
This number is so large, it's almost impossible to describe in human terms.  It isn't sufficient to say &quot;the number has this many digits&quot; because it has more digits than particles in the universe.  In fact, the number of digits this number contains has more digits than the number of particles in the universe.
Let this number be x, and let f(0) = x.  Recursively define f(n) = floor[log10[f(n-1)]].  The first n which is smaller than 1 googol is over 10,000.
This answer is 93 bytes.
If we removed removed a single leading +, the result would be 65536 = 2222.  Removing another + gives us 4.  This means that if we add another +, we would get a number tower with 2↑↑↑4 twos, which would be 2↑↑↑5.  That answer would be 94 bytes.  If I were to add another 406 plus signs to this, we would get 2↑↑↑411, which is mind-bogglingly big.  Instead, if  I were to add yet another outer loop, I could probably get something close to 2↑↑↑↑380.  For the record, 2↑↑↑↑3 = 2↑↑↑4, and 2↑↑↑↑4 dwarfs 2↑↑↑4 much the same way that 2↑↑↑4 dwarfs 65536.  In fact, it may just be possible to actually get close to Graham's Number using 500 bytes of brainfuck.