3 Don't edit my posts.
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dc, 12 bytes

Uses an "anonymous" recursive macro, should be able to run for ever:

z[r1+prdx]dx

dc, 11 bytes (not valid)

Fails the requirements as it begins from 0 (not 1) and it uses the stack depth as a counter (invalid because "you cannot store a unary representation of the current number in memory"; it will run out of memory sooner or later):

[zplxx]dsxx

dc, 12 bytes

Uses an "anonymous" recursive macro, should be able to run for ever:

z[r1+prdx]dx

dc, 11 bytes (not valid)

Fails the requirements as it begins from 0 (not 1) and it uses the stack depth as a counter (invalid because "you cannot store a unary representation of the current number in memory"; it will run out of memory sooner or later):

[zplxx]dsxx

dc, 12 bytes

Uses an "anonymous" recursive macro, should be able to run for ever:

z[r1+prdx]dx
2 Reorder with the valid solution first (to give correct leaderboard result). Incorporate comments into answer.
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dc, 1112 bytes

Uses the stack depth as a counteran "anonymous" recursive macro, so you willshould be able to run out of memory sooner or laterfor ever:

[zplxx]dsxxz[r1+prdx]dx

dc, 1211 bytes (not valid)

Uses an "anonymous" recursive macro, should be able toFails the requirements as it begins from 0 (not 1) and it uses the stack depth as a counter (invalid because "you cannot store a unary representation of the current number in memory"; it will run for everout of memory sooner or later):

z[r1+prdx]dx[zplxx]dsxx

dc, 11 bytes

Uses the stack depth as a counter, so you will run out of memory sooner or later:

[zplxx]dsxx

dc, 12 bytes

Uses an "anonymous" recursive macro, should be able to run for ever:

z[r1+prdx]dx

dc, 12 bytes

Uses an "anonymous" recursive macro, should be able to run for ever:

z[r1+prdx]dx

dc, 11 bytes (not valid)

Fails the requirements as it begins from 0 (not 1) and it uses the stack depth as a counter (invalid because "you cannot store a unary representation of the current number in memory"; it will run out of memory sooner or later):

[zplxx]dsxx
1
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dc, 11 bytes

Uses the stack depth as a counter, so you will run out of memory sooner or later:

[zplxx]dsxx

dc, 12 bytes

Uses an "anonymous" recursive macro, should be able to run for ever:

z[r1+prdx]dx