Count Up, Replace, Repeat!

Jelly, 3 bytes

R¡F

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Explanation

R¡F    Argument n

R      Yield range [1..n]
 ¡     Repeat n times
  F    Flatten the result

Python, 50 bytes

lambda i:eval("[i "+"for i in range(1,i+1)"*i+"]")

Scope abuse! For example, for i=3, the string to be evaluated expands to.

[i for i in range(1,i+1)for i in range(1,i+1)for i in range(1,i+1)]

Somehow, despite using the function input variable i for everything, Python distinguishes each iteration index as belonging to a separate scope as if the expression were

[l for j in range(1,i+1)for k in range(1,j+1)for l in range(1,k+1)]

with i the input to the function.

05AB1E, 6 3 bytes

DFL

Explained

D     # duplicate input
 F    # input times do
  L   # range(1,N)

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Saved 3 bytes thanks to @Adnan