Does a spell with replicate go on the stack once or multiple times?

When you cast a spell with replicate, the original spell is placed on the stack exactly once, just like any other spell (CR 601.2). Replicate is a triggered ability that checks how many times you paid the replicate cost as an additional cost while casting the spell.

For each time you paid the replicate cost, replicate triggers and puts a copy of the spell onto the stack (CR 702.56b). These copies are not cast — they are created directly on the stack — and they do not trigger replicate again. Each copy can be targeted independently if the spell requires targets.

The copies resolve separately, one at a time, in last-in-first-out order (CR 608.2). The original spell is also still on the stack and resolves as well, so paying replicate twice means three total instances of the spell effect: two copies plus the original.

Example: You cast Train of Thought (draw cards equal to the number of times it was kicked… actually a replicate card like Pyromatics) paying its replicate cost twice. One original spell is placed on the stack. Replicate triggers twice, placing two copies onto the stack. You now have three instances on the stack — your opponent can respond to each separately as they resolve.

In summary: one casting goes on the stack, but each replicate payment generates one additional copy via a triggered ability, giving you one original plus one copy per replicate payment (CR 702.56a–702.56b).