Does the Arena's Greatest battlefield perform a check at the beginning of every beginning phase that can be reacted to?

Yes, the triggered ability of The Arena's Greatest creates a chain at the start of the specified Beginning Phase, which allows players to respond with [Reaction] cards or abilities.

According to Rule 383.3 and supported by RiftJudge FAQ #9468, when a triggered ability's condition is met, it is placed on the chain, creating a Closed State. This allows players to play [Reaction] cards or activate [Reaction] abilities in response to the trigger before it resolves.

Sequence of Events:

1. Trigger: At the start of the turn player's first Beginning Phase, the condition for The Arena's Greatest is checked. Since the condition is met, the ability triggers and is added to the chain.
2. Priority: Once the ability is on the chain, a Closed State is created, and priority passes to the players.
3. Reaction: Players may play [Reaction] cards or activate [Reaction] abilities.
4. Resolution: The chain resolves in reverse order (LIFO - Last In, First Out). If a reaction changes the game state such that the ability no longer has a legal target or valid conditions (if applicable), the effect would fail to resolve as intended.

Note on Frequency: Per the card text ("At the start of each player's first Beginning Phase"), this trigger only occurs once per player per game. Once a player has had their first Beginning Phase of the game, this specific ability will not trigger for them again in that game. (Reference: RiftJudge FAQ #506.3.c)