pure Bash, (削除) 116 (削除ここまで) 112 bytes

Based on answers from iBug and Léa Gris

#!/bin/bash
test116()
{
n=1ドル;shopt -s extglob;while((n>9));do m=0;for x in ${n//?()/ };{ ((m<x))&&m=$x;};n=$[10#${n/$m/}*m];done;echo $n
}
for k in 9 27 757 1234 26364 432969 1234584 91273716; do printf '%d -> ' "$k"; test116 "$k"; done

Try it online!

Output

Explained:

n=1ドル; # (*) get argument n
shopt -s extglob; # enable ${n//?()/ } below
while((n>9));do # loop until number is <=9
 m=0; # preset maximum to 0
 for x in ${n//?()/ };do # loop over the digits
 ((m<x))&&m=$x; # find max
 done;
 n=$[ # calculate new number
 10# # force base 10 as 0N becomes octal
 ${n/$m/} # remove first occurance of max
 *m]; # multiply m
done;
echo $n # output result

• Shebang #!/bin/bash+LF (12 byte) is not included in the count.
  • We are in bash, so this is already the default
• If number can directly be passed in as n, 5 bytes (marked with (*) above) can be shoved away
  • Instead ./golf.sh 1234 it would look like n=1234 ./golf.sh
  • However, shell scripts usually get data passed in via argument
  • A read n; if number is passed in from STDIN would add 2 byte.
• shopt -s extglob; can be left away, if this option is given on commandline
  • Like in n=1234 bash -Oextglob ./golf.sh
  • But extglob is not set by default, hence it is included in the count
• 10# is needed as ${n/$m/} can leave a number with a leading 0
  • Without the 10# input 91273716 wrongly returns 8.
• Tested on bash 5.0.17(1)-release from bash 5.0-6ubuntu1.1 from Ubuntu 20.04 LTS

pure Bash, (削除) 116 (削除ここまで) 112 bytes, or 101, or even 97?

Based on answers from iBug and Léa Gris and tips-for-golfing-in-bash (thanks, @AaronMiller and @BrowncatPrograms!):

#!/bin/bash
test112()
{
n=1ドル;shopt -s extglob;while((n>9));do m=0;for x in ${n//?()/ };{ ((m<x))&&m=$x;};n=$[10#${n/$m/}*m];done;echo $n
}
for k in 9 27 757 1234 26364 432969 1234584 91273716; do printf '%d -> ' "$k"; test112 "$k"; done

Try it online!

Also there are two shorter variants:

#!/bin/bash
f()(shopt -s extglob;((1ドル>9))||exit 1ドル;m=0;for x in ${1//?()/ };{ ((m<x))&&m=$x;};f $[10#${1/$m/}*m])
for k in 9 27 757 1234 26364 432969 1234584 91273716; do printf '%d -> ' "$k"; f "$k"; echo "$?"; done

(Try it online! ) with 101 bytes and with 97 bytes (Try it online! ):

shopt -s extglob;((1ドル>9))||exit 1ドル;m=0;for x in ${1//?()/ };{ ((m<x))&&m=$x;};0ドル $[10#${1/$m/}*m]

But I am not entirely sure the last two apply to the rules. The latter even uses 0ドル (I neither found a rule allowing that nor denying this) to recurse to itself, so it only works if put into a script (like golf.sh), made executable and then is called like golf.sh 91273716; echo $?

Output of the scripts:

Explained first variant (112 bytes):

n=1ドル; # (*) get argument n
shopt -s extglob; # enable ${n//?()/ } below
while((n>9));do # loop until number is <=9
 m=0; # preset maximum to 0
 for x in ${n//?()/ };{  # loop over the digits
 ((m<x))&&m=$x; # find max
 };
 n=$[ # calculate new number
 10# # force base 10 as 0N becomes octal
 ${n/$m/} # remove first occurance of max
 *m]; # multiply m
done;
echo $n # output result

• Shebang #!/bin/bash+LF (12 byte) is not included in the count.
  • We are in bash, so this is already the default
• AFAICS passing in the number directly as variable n (see marked (*) above) is not allowed. Hence these 5 bytes are needed.
  • Else ./golf.sh 1234 would look like n=1234 ./golf.sh
  • A read n; if number is passed in from STDIN is 2 byte longer
• shopt -s extglob; can be left away, if this option is given on commandline
  • Like in n=1234 bash -Oextglob ./golf.sh
  • But extglob is not set by default, hence AFAICS it must be included in the count
• 10# is needed as ${n/$m/} can leave a number with a leading 0
  • Without the 10# input 91273716 wrongly returns 8.

Explained shortest variant (97 bytes):

shopt -s extglob; # see above
((1ドル>9))||exit 1ドル; # stop if goal reached
m=0;for x in ${1//?()/ };{ ((m<x))&&m=$x;}; # max calc, as above
0ドル $[10#${1/$m/}*m] # recurse to next step

Notes:

• Actually the last line is a tail recursion, so bash uses exec and does not fork

pure Bash, 116 bytes

#!/bin/bash
test116()
{
n=1ドル;shopt -s extglob;while((n>9));do m=0;for x in ${n//?()/ };do ((m<x))&&m=$x;done;n=$[10#${n/$m/}*m];done;echo $n
}
for k in 9 27 757 1234 26364 432969 1234584 91273716; do printf '%d -> ' "$k"; test116 "$k"; done

Try it online!

pure Bash, (削除) 116 (削除ここまで) 112 bytes

#!/bin/bash
test116()
{
n=1ドル;shopt -s extglob;while((n>9));do m=0;for x in ${n//?()/ };{ ((m<x))&&m=$x;};n=$[10#${n/$m/}*m];done;echo $n
}
for k in 9 27 757 1234 26364 432969 1234584 91273716; do printf '%d -> ' "$k"; test116 "$k"; done

Try it online!

pure Bash, 116 bytes

pure Bash, 116 bytes