the highest number a binary system can display is one less than 2 to the power the number of bits available to it.
So a 4 bit system can display a highest number of 15. This is 2 to the power of 4 = 2 x 2 x 2 x 2 = 16; less one is 15.
In binary terms this is easy to understand. You have four bits, each of which can display 1 or 0:
0000 = 0
0001 = 1
0010 = 2
0011 = 3
0100 = 4
0101 = 5
0110 = 6
0111 = 7
1000 = 8
1001 = 9
1010 = 10
1011 = 11
1100 = 12
1101 = 13
1110 = 14
1111 = 15
and now you've run out of bits to display numbers
however, these numbers are not signed - signing meaning positive or negative. Since one of the bits becomes the sign, the maximum signed number a binary system can display is one less than 2 to the power of one less than the number of bits available to it.
So a 4 bit system can display a highest signed number of +7. This is 2 to the power of (4-1) = 2 x 2 x 2 = 8; less one is 7. Again, let's speak to the binary:
+000 = 0
+001 = +1
+010 = +2
+011 = +3
+100 = +4
+101 = +5
+110 = +6
+111 = +7
and we've run out again. +8 as a signed integer makes no sense to a 4 bit system and it'll die as a result.
An 8 bit system can thus display up to ((2^8)-1) - 255 - as a valid number and up to ((2^(8-1)-1) - +127 - as a valid signed number. You'll notice we've only doubled the bits available but the integers have increased by a multiple of 16.
Double it again to 16 bits and now it can display ((2^16)-1) - 65,535 - as a valid number and up to ((2^(16-1)-1) - +32,767 - as a valid signed number. Again, all we've done is double the number of bits available, but the integers available have increase by a multiple of 256.
A 32 bit system like ps2 can display up to ((2^32)-1) - 4,294,967,295 - as a valid number and up to ((2^(32-1)-1) - +2,147,483,647 - as a valid signed number. This latter one should be very familiar to gran turismo fans as the maximum speed of a wheelie in a suzuki escudo in gt3 before the game locked up. And it locked up for precisely this reason - trying to display a number bigger than what it had bits available to make sense to it.
The ps3 is a 64 bit system, so its unsigned and unsigned limits are ((2^64)-1) - 18,446,744,073,709,551,615 - and ((2^(64-1)-1) - +9,223,372,036,854,775,807.
So, assuming the day count in gt5 is a signed integer (if it wasn't it ought to behave quite oddly, progressing from a positive day to a negative day because it didn't know the difference), you'll never be able to reach day 9,223,372,036,854,775,808 because the console would lock up. And it's academic, because even if you could skip a day per second it'd take you 300 billion years to get there.