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What is the maximum amount of memory supported by a dual core machine?

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Volatile or Non-Volatile Memory?

If it is Volatile, like RAM, then it isn't by the number of cores, but by the number of physical slots available and by processor Bit-Rate. 32 Bit Systems (Processors, Operating Systems, or both) can only access up to 4~ GB of Volatile memory, such as RAM. This is lowered to 3GB, since there is an overhead associated with background operations.
Calculation:
2^32 = 4,294,967,296 bytes
4,294,967,296 / (1,024 x 1,024) = 4,096 MB = 4 Gigabytes if all things are perfect. 64 bit Systems (Processors, Operating Systems, or both) can access a MUCH, MUCH larger well of Volatile memory. Something along the lines of EXAbytes. (Which is two steps beyond Terabyte) 2^64 = 18,446,744,073,709,551,616 bytes
18,446,744,073,709,551,616 / (1,024(Kb) x 1,024(Mb) x 1,024(Gb) x 1,024(Tb)) = 16EB (Exabytes)

Many 64-bit processors have 48-bit addressing, so their actual maximum memory addressing capability is in the Petabyte range.

Non-volatile memory addressing is essentially limitless. I believe there are theoretical limitations, but I don't have the knowledge at hand to calculate them.
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What is the maximum amount of memory for a dual core machine?

Most modern processors are 64 bit, so they're capable of addressingridiculously large amounts of memory if the OS is also 64-bit ...the practical limits are really "what sizes