Rayo's number is an incredibly large number, and the function it's built upon grows so fast it's uncomputable. If you're on this page, there's a good chance you already know about it. If you don't, you can click the link to read about it.
Anyway, since its original definition in 2007 people have been hard at work determining the size of smaller values of the Rayo function. Of note is a considerable resurgence in efforts starting in 2020, resulting in a series of lower bounds for Rayo(n) based on various values of n.
Beyond n=320 the values begin to rise rapidly thanks to the fantastic lower bound of Rayo(260 + 20n) > 2 ^^ n, meaning that Rayo(340) > 65,537 and Rayo(360) > 10^19,728. These values do not fit on the graph.
Rayo(n) >= N/A
Here's a quick (and slightly off) history of the development of lower bounds for the function:
26 January 2007 - Rayo's number is defined.
< 5 February 2013 - Rayo(10) is shown to be >= 0 using "(¬∃1(1∈2))".
< 5 February 2013 - Rayo(38) is shown to be >= 1 using "(((¬∃3(3∈2))∧2∈1)∧(¬∃3((3∈1∧(¬3=2)))))".
10 August 2016 - The lower bound Rayo((9n^2+47n+20)/2) >= n is defined using a continuation of the earlier examples (modified). This puts Rayo(1000) >= 12.
22 April 2020 - The lower bound Rayo(835+96n) > 2^^n is defined.
23 April 2020 - The above formula is improved to Rayo(768+84n) > 2^^n.
24 April 2020 - The above formula is improved again to Rayo(733+65n) > 2^^n. This puts Rayo(1000) > 65,536.
30 May 2020 - Rayo(7901) is shown to be > S(2^65536 - 1) where S(n) is the maximum shifts function.
19 July 2020 - Rayo(n) is proven to be 0 for n < 10.
26 July 2020 - The above formula is improved again to Rayo(294+74n) > 2^^n. This puts Rayo(1000) > 10^^6.
< 1 August 2020 - The above formula is improved again to Rayo(362+20n) > 2^^n. This puts Rayo(1000) > 10^^28.
19 August 2020 - The above formula is improved again to Rayo(266+20n) > 2^^n. This puts Rayo(1000) > 10^^33.
15 May 2022 - The above formula is improved again to Rayo(260+20n) > 2^^n. This puts Rayo(1000) > 10^^34. With this formula, Rayo(7339) is known to be > S(2^65536 - 1).