Infinity is literally defined as a number greater than any assignable quantity or countable number. If you can define it with a real number, by definition it can't be infinity (countable or uncountable), since you could simply add 1 to that number to make it bigger. Some infinities are greater than others, but they're all bigger than 10103.
10103=10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 (a one with 103 zeros after it)
The precision of the estimate is built into the number if you understand scientific notation. It's only got one significant figure, meaning it could be out by up to 50%. We have very strong evidence that we're not in thermodynamic equilibrium, but we rely on estimates for the exact figure because we're limited to data from the observable universe.