Confusing Alphabet Staircase

Jelly, 9 bytes

ØAjṪ$Ƥż¹Y

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How it works

ØAjṪ$Ƥż¹Y  Main link. No arguments.

ØA         Yield "ABCDEFGHIJKLMNOPQRSTUVWXYZ".
     Ƥ     Map the link to the left over all prefixes, i.e., ["A", "AB", ...].
    $        Combine the two links to the left into a chain.
   Ṫ           Tail; yield and remove the last letter of each prefix.
  j            Join the remainder, using that letter as separator.
      ż¹   Zip the resulting strings and the letters of the alphabet.
        Y  Separate the results by linefeeds.

C, 82 bytes

f(i,j){for(i=!puts("A");++i<26;puts(""))for(j=0;j++<i*2;)putchar(65+(j&1?j/2:i));}

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R, 50 bytes

l=LETTERS
for(i in 0:25)cat(l[0:i],"
",sep=l[i+1])

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Perhaps the cleverest part here is using letters[0] for the empty string to get cat(character(0),'\n',sep="A") to print the first line.