And we are not talking about hypothetical large numbers that do not represent anything. So googol [10^100], googolplex [10^googol], Graham's number, and Moser's number don't count.
The biggest number that actually represents something can be found right inside our brains, literally. Which brain contains about 10^27 atoms incidentally. In John D. Barrow's The Constants of Nature (2002), we find the following discussion [pp. 116-118]:
"... astronomy is not the place to look. The big numbers of astronomy are additive. They arise because we are counting stars, planets, atoms and photons in a huge volume. If you want really huge numbers you need to find a place where the possibilities multiply rather than add. For this you need complexity. And for complexity you need biology.
"In the seventeenth century the English physicist Robert Hooke [1635-1703] made a calculation 'of the number of separate ideas the mind is capable of entertaining' . The answer he got was 3,155,760,000. Large as this number might appear to be (you would not live long enough to count up to it!) it would now be seen as a staggering underestimation. Our brains contains about 10 billion neurons, each of which sends out feelers, or axons, to link it to about one thousand others. These connections play some role in creating our thoughts and memories. How this is done is still one of Nature's closely guarded secrets. Mike Holderness suggests that one way of estimating  the number of possible thoughts that a brain could conceive is to count all those connections. The brain can do many things at once so we could view it as some number, say a thousand, little groups of neurons. If each neuron makes a thousand different links to the ten million others in the same [neuron] group then the number of different ways in which it could make connections in the same neuron group is 10^7 x 10^7 x 10^7 x
... one thousand times. This gives 10^7000 possible patterns of connections. But this is just the number for one neuron group. The total number for 10^7 neurons is 10^7000 multiplied together by 10^7 times. This is 10^70,000,000,000. If the 1000 or so groups of neurons can operate independently of each other then each of them contributes 10^70,000,000,000 possible wirings, increasing the total to the Holderness number, 10^70,000,000,000,000.
"This is the modern estimate of the number of different electrical patterns that the brain could hold. In some sense it is the number of different possible thoughts or ideas that a human brain could."
Hence for now, it looks like the Holderness number is it. There's still no Wikipedia entry on the Holderness number incidentally.
For more on large numbers, please see also Scott Aaronson's "Who Can Name the Bigger Number?"
 The estimate was reported in Albrecht von Haller's Elementa Physiologiae, vol. 5, London, 1786, p. 547.
 The estimates were made by Mike Holderness, in 'Think of a Number', New Scientist, 16 June 2001, p. 45."
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