#Java 8, (削除) 126 (削除ここまで) 104 bytes

n->{for(;n>9;n=new Long((n+"").replaceFirst((n=(n+"").chars().max().getAsInt()-48)+"",""))*n);return n;}

-22 bytes thanks to @OlivierGrégoire.

Explanation:

Try it here.

n->{ // Method with long as both parameter and return-type
 for(;n>9; // Loop as long as the number contains more than 1 digit
 n= // Replace the current number with:
 new Long((n+"").replaceFirst((n=(n+"").chars().max().getAsInt()-48)+"",""))
 // Remove the first largest digit from the number,
 *n // and multiply this new number with the removed digit
 ); // End of loop
 return n; // Return the result
} // End of method

Java 8, (削除) 126 (削除ここまで) 104 bytes

n->{for(;n>9;n=new Long((n+"").replaceFirst((n=(n+"").chars().max().getAsInt()-48)+"",""))*n);return n;}

-22 bytes thanks to @OlivierGrégoire.

Explanation:

Try it here.

n->{ // Method with long as both parameter and return-type
 for(;n>9; // Loop as long as the number contains more than 1 digit
 n= // Replace the current number with:
 new Long((n+"").replaceFirst((n=(n+"").chars().max().getAsInt()-48)+"",""))
 // Remove the first largest digit from the number,
 *n // and multiply this new number with the removed digit
 ); // End of loop
 return n; // Return the result
} // End of method

#Java 8, 126 bytes

long c(long n){int m=0;for(int c:(n+"").getBytes())m=c-48>m?c-48:m;return n<10?n:c(m*new Long((n+"").replaceFirst(m+"","")));}

Explanation:

Try it here.

long c(long n){ // Method with long as both parameter and return-type
 int m=0;  // Max-digit, starting at 0
 for(int c:(n+"").getBytes()) // Loop over the digits of the input
  m=c-48>m?  // If the current digit is larger than the current max:
 c-48 // Use the current digit as new max
 :  // Else:
 m;  // Leave the max digit the same
 // End of loop (implicit / single-line body)
 return n<10?  // If the input is a single digit:
 n  // Return this digit as result
 :  // Else:
 c(m* // Recursive-call with this max digit, multiplied by
 new Long((n+"").replaceFirst(m+"",""))); 
 // the input minus this first digit
} // End of method

#Java 8, (削除) 126 (削除ここまで) 104 bytes

n->{for(;n>9;n=new Long((n+"").replaceFirst((n=(n+"").chars().max().getAsInt()-48)+"",""))*n);return n;}

-22 bytes thanks to @OlivierGrégoire.

Explanation:

Try it here.

n->{ // Method with long as both parameter and return-type
 for(;n>9; // Loop as long as the number contains more than 1 digit
 n= // Replace the current number with:
 new Long((n+"").replaceFirst((n=(n+"").chars().max().getAsInt()-48)+"",""))
 // Remove the first largest digit from the number,
 *n // and multiply this new number with the removed digit
 ); // End of loop
 return n; // Return the result
} // End of method