Traveling Chocobo

Replacement effects are unaffected by Traveling Chocobo's ability. For example, a Bird that enters the battlefield under your control with one +1/+1 counter on it won't receive an additional +1/+1 counter.

In a Two-Headed Giant game, Traveling Chocobo's second ability does not affect triggered abilities of permanents your teammate controls.

Traveling Chocobo's ability doesn't copy the triggered ability; it just causes the ability to trigger an additional time. Any choices made as you put the ability onto the stack, such as modes and targets, are made separately for each instance of the ability. Any choices made on resolution, such as whether to put counters on a permanent, are also made individually.

If a triggered ability is linked to a second ability, additional instances of that triggered ability are also linked to that second ability. If the second ability refers to "the exiled card," it refers to all cards exiled by instances of the triggered ability.

The top card of your library isn't in your hand, so you can't cycle it, discard it, or activate any of its abilities that could be activated from your hand.

If you control two Traveling Chocobos, a land or Bird you control entering causes abilities to trigger three times, not four. A third Traveling Chocobo causes abilities to trigger four times, a fourth causes abilities to trigger five times, and so on.