Garlandification

Python, 60 bytes

f=lambda s,i=1:s.find(s[i:])and f(s,i+1)or(len(s)-i)*s[:i]+s

Was hoping for better, but oh well. s.find works neatly here in place of not s.startswith.

Retina, 58 bytes

.+
$0#$0
(.*)(.+)#.*\1$
$0#$1#$2-
+`\w#(\w*)-
#$1-$1
#.*-
<empty line>

Each line should go to its own file but you can run the code as one file with the -s flag.

The four substitution pairs do the followings:

• Duplicate word so we can search for overlaps too.
• Append the word split up at order number of characters.
• Append the last part order times.
• Keep the original word and the lastly appended part and drop everything else.

The string states for the example onion:

onion
onion#onion
onion#onion#on#ion-
onion#onion##ion-ionion
onionionion

Pyth, 19 18 bytes

+z*>Kf!xz>zT1zl>zK

Try it online: Demonstration or Test harness

Explanations:

+z*>Kf!xz>zT1zl>zK   implicit: z = input string
     f      1        find the first number T >= 1, which satisfies:
         >zT            all but the first T chars of z
       xz               index of ^ in z
      !                 == 0
    K                store in K
                     the order is length(z) - K
   >K        z       the last K chars
  *                  repeated
              l>zK   len(all but the last K chars) times
+z                   insert z at the beginning