107 104 94 Bytes
How it works:
Stores 26 on the second cell, 65 on the third cell, 10 on the sixth cell, and 1 on the fourth, leaving the pointer on the fourth (Saved
3 13 bytes by making 26, 10, and 97 65 at the same time).
Moves the pointer back to the start.
Now we're ready to get started!
[ | While the first cell is non-zero (so 26 times)
>> | Move to cell 3
[ | While cell 3 is non-zero
<.>>+<- | Print cell 2, increment cell 4, and decrement cell 3
>>.<< | Move to cell 5 and print it (new line), then return
<+>+> | Increment cells 2 and 3, and move to 4
[<+>-] | Add the value at 4 to 3
<<<- | Decrement cell 1
The important part is that cell 3 stores the number of times to print a letter. Each time a letter is printed, the value at 3 is moved to 4 and increased by 1, so that the next number is printed one more time than the last.
I'm completely new to brainfuck, so I wouldn't be surprised if this could be improved. However, I was also surprised by how fun brainfuck is to write, as well as by how easy it was to write, when compared with what I had imagined.
I would definitely recommend learning brainfuck to any bored programmers out there ;)