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##Lua, 154 Bytes

I should have some ways to golf this down, I'm experimenting right now.

n=...z=table
while n+0>9 do
t={}T={}n=n..''n:gsub(".",function(c)t[#t+1]=c T[#T+1]=c
end)z.sort(t)x=t[#t]z.remove(T,n:find(x))n=z.concat(T)*x
end
print(n)

Try it online!

###Explanations

n=... -- define n as a shorthand for the argument
z=table -- define z as a pointer to the object table
while n+0>9 -- iterate as long as n is greater than 9
do -- n+0 ensure that we're using a number to do the comparison
 t={} -- intialise two tables, one is used to find the greatest digit
 T={} -- the other one is used to remove it from the string
 n=n..'' -- ensure that n is a string (mandatory after the first loop)
 n:gsub(".",function(c) -- apply an anonymous function to each character in n
 t[#t+1]=c -- fill our tables with the digits
 T[#T+1]=c
 end) 
 z.sort(t) -- sort t to put the greatest digit in the last index
 x=t[#t] -- intialise x to the value of the greatest digit
 z.remove(T,n:find(x)) -- remove the first occurence of x from the table T 
 -- based on its position in the input string
 n=z.concat(T)*x -- assign the new value to n
end -- if it still isn't a single digit, we're looping over again
print(n) -- output the answer