Lengthen letter runs

Python 3, 44 bytes

for c in input():print(c,end=c[id==c:]);id=c

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05AB1E, 5 bytes

.¡€ĆJ

Explanation:

Example input: "dddogg"
.¡       Split into chunks of consecutive equal elements
         stack: [['ddd', 'o', 'gg']]
  €      For each...
   Ć       Enclose; append the first character to the end of the string
         stack: [['dddd', 'oo', 'ggg']]
    J    Join array elements into one string
         stack: ['ddddooggg']
Implicitly output top element in stack: "ddddooggg"

Try it online or as a test suite.

Enclose is a pretty new builtin; it's the first time I've used it. Very convenient ;)

05AB1E, 4 bytes (non-competing)

γ€ĆJ

.¡ has been replaced by γ in the latest update.

Retina, 11 bytes

(.)\1*
$1$&

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Replaces each run of characters with one of the run's characters followed by the run itself.