05AB1E, 8 bytes

×ばつì}J

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Explanation

×ばつì}J Arguments s, n ("Hello, World!", 3)
ô Split s into pieces of n (["Hel", "lo,", ...])
 ε } Apply each
 ć Extract head ("Hel" -> "el", "H" ...)
 ×ばつì Repeat n times and prepend ("el", "H" -> "HHHel" ...)
 J Join stack  ("HHHellllo, Worrrld!!!")

05AB1E, (削除) 8 (削除ここまで) 7 bytes

-1 byte thanks to @Emigna

×ばつì?

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Explanation

×ばつì? Arguments s, n ("Hello, World!", 3)
ô Split s into pieces of n (["Hel", "lo,", ...])
 ʒ Filter (used as foreach)
 ć Extract head ("Hel" -> "el", "H" ...)
 ×ばつì Repeat n times and prepend ("el", "H" -> "HHHel" ...)
 ? Print without newline

05AB1E, 8 bytes

×ばつì}J

Try it online!

Explanation

×ばつì}J Arguments s, n ("Hello, World!", 3)
ô Split s into pieces of n (["Hel", "lo,", ...])
 ε } Apply each
 ć Extract head ("Hel" -> "el", "H" ...)
 ×ばつì Repeat n times and prepend ("el", "H" -> "HHHel" ...)
 J Join stack ("HHHellllo, Worrrld!!!")

05AB1E, 8 bytes

×ばつì}J

Try it online!

Explanation

×ばつì}J Arguments s, n ("Hello, World!", 3)
ô Split s into pieces of n (["Hel", "lo,", ...])
 ε } Apply each
 ć Extract head ("Hel" -> "el", "H" ...)
 ×ばつì Repeat n times and prepend ("el", "H" -> "HHHel" ...)
 J Join stack ("HHHellllo, Worrrld!!!")