Alphabet Staircase

Python 2, 36 bytes

i=1
exec'print chr(i+96)*i;i+=1;'*26

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05AB1E, 2 bytes

Aƶ

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Note that this outputs as a list of lines, as the OP explicitly allowed. The link uses a version with pretty-print (joined by newlines).

How it works

• A yields the lowercase alphabet.
• ƶ lifts the alphabet (multiplies each element by its index).
• » joins by newlines.

APL (Dyalog), 12 8 5 bytes ^SBCS

3 bytes saved thanks to @ngn

4 bytes saved thanks to @Adám

⍴⍨⌸⎕A

OP clarified uppercase letters are valid, as well as output as an array of strings.

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How?

⌸ gives us every letter in the ⎕A lphabet with its indexes in it, handed into the function ⍴⍨ with the letter as left argument and the indexes as right argument.

⍴⍨ resha ⍴ es its right argument to the length supplied by its left one. ⍨ switches the left and right (therefore the symbol of it, looking like the face of someone reading this explanation).