The middle-growing hierarchy is a hierarchy created by Googology Wiki user Ikosarakt1.

The rules are as following:

• $$m(0,n)=n+1$$
• $$m(\alpha+1,n)=m(\alpha,m(\alpha,n))$$
• $$m(\alpha,n)=m(\alpha[n],n)$$

Although it is not clarified in the original definition, $$\alpha$$ denotes a countable ordinal equipped with a fixed system of fundamental sequences of limit ordinals up to $$\alpha$$, and $$n$$ denotes a natural number.

We can see that the only difference between the middle-growing hierarchy and the fast-growing hierarchy is that $$m_{\alpha+1}(n) = m_{\alpha}^2(n)$$ while $$f_{\alpha+1}(n) = f_{\alpha}^n(n)$$, where $$m_{\alpha}(n)$$ denotes $$m(\alpha,n)$$ and the superscripts $$^2$$ and $$^n$$ represent an iteration of the functions $$m_{\alpha}$$ and $$f_{\alpha}$$.

## Up to $$\omega^\omega$$

\begin{eqnarray*} m(0,n) &=& n + 1 \\ m(1,n) &=& n + 2 \\ m(2,n) &=& n + 4 \\ m(3,n)  &=& n + 8 \\ m(k,n) &=& n + 2^k \\ m(\omega,n) &=& n + 2^n \\ m(\omega+1,n) &=& n + 2^n + 2^{n + 2^n} \\ m(\omega+2,n) &=& n + 2^n + 2^{n + 2^n} + 2^{n + 2^n + 2^{n + 2^n}} > 2^{2^{2^n}} \\ m(\omega+m,n) &>& \textrm En\#(m+1) > 2\uparrow\uparrow(m+1) \\ m(\omega2,n) &>& 2\uparrow\uparrow(n+1) \\ m(\omega3,n) &>& 2\uparrow\uparrow\uparrow(2^n) \\ m(\omega m,n) &>&  2\uparrow^m(2^n) \\ m(\omega^2,n)  &>& 2\uparrow^n(2^n) \\ m(\omega^2+\omega,n)  &>& \lbrace n,2^n,1,2 \rbrace \\ m(\omega^22,n)  &>& \lbrace n,2^n,n,2 \rbrace \\ m(\omega^3,n) &>&  \lbrace n,2^n,n,n \rbrace \\ m(\omega^m,n) &>& \lbrace n,m+1 (1) 2 \rbrace \\ m(\omega^{\omega},n) &>& \lbrace n,n+1 (1) 2 \rbrace > \lbrace n,n (1) 2 \rbrace \end{eqnarray*}

We see that the middle-growing hierarchy catches the fast-growing hierarchy at $$\omega^{\omega}$$, and generally, it does so at all multiples of it.

## Specific numbers

135 is equal to m(ω, 7), and also the number of nominal AM radio frequencies (n × 9 kHz), where 17 ≤ n ≤ 31 or 59 ≤ n ≤ 178) in Europe, and the bandwidth of the longwave radio band (in kHz).

264 is equal to m(ω, 8), and also approximately the number of U.S. gallons in a cubic metre, and the highest possible game value (Grand ouvert with all four jacks) in the German card game of Skat.
{"ft_lang_label":"__label__en","ft_lang_prob":0.5224861,"math_prob":0.9999913,"size":4318,"snap":"2022-27-2022-33","text_gpt3_token_len":1430,"char_repetition_ratio":0.17709783,"word_repetition_ratio":0.015151516,"special_character_ratio":0.35919407,"punctuation_ratio":0.097428955,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9999666,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-06-26T12:03:48Z\",\"WARC-Record-ID\":\"<urn:uuid:444d18e1-5f4c-4fcc-911e-7efee7785cba>\",\"Content-Length\":\"367482\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:d68b5af6-9d91-431b-908e-ff7a5307fa4c>\",\"WARC-Concurrent-To\":\"<urn:uuid:65c3df49-9046-4d8a-866a-5d63e31369a8>\",\"WARC-IP-Address\":\"151.101.0.194\",\"WARC-Target-URI\":\"https://googology.fandom.com/wiki/Middle-growing_hierarchy\",\"WARC-Payload-Digest\":\"sha1:ZTMUJ2DQN5WMGXQNDOCTWI7LR2BBPPML\",\"WARC-Block-Digest\":\"sha1:4SIX4USC7XZFQCSLT7B2YHA4BVJ5J4VM\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656103205617.12_warc_CC-MAIN-20220626101442-20220626131442-00089.warc.gz\"}"}