If you took one second for every 10 numbers (this is impossible for larger numbers) it would take over 19,000 years to count to 6 trillion.

To count to 6 trillion in 6 hours you would need to count 277,777,777 numbers per second.

Numbers can be so big that they cannot be written down using every atom in the universe. Using Knuth's up arrow notation numbers can be very large. 9↑↑10 means 9^(9^(9^(9^(9^(9^(9^(9^(9^9)))))))) this number is impossible to write out the digits in fact it is much larger than the number of atoms in the universe. Even it's number of digits is larger than the atoms in the universe.

Then you can have 3↑↑↑↑3 this is more complicated.

Graham's number is a incomprehensibly large number. Using knuth's up arrow notation it is:

g₁=3↑↑↑↑3

g₂=3↑↑↑↑..(g₁ arrows)..↑↑↑↑3

g₃=3↑↑↑↑..(g₂ arrows)..↑↑↑↑3

This goes on upto g₆₄ which is Graham's number.

Edit:

Even counting by tens you would still need to count about 27,777,777 numbers per second for six hours. So you are saying you counted over 27 and 3 quarter million numbers per second are you? If you counted in 30,000,000's it may be possible. You must have counted to a different number or missed many numbers out. I could count 8-10 numbers per second untill I get to the larger numbers but nobody can count 27.78 million numbers per second. Either that or you are lying because it is impossible to count to 6 trillion even counting in tens. Perhaps then your work's system shutdown for 1,900 Earth years and one night lasts 3,800 Earth years where you live.