Replacement Ability Question

[ISBPathfinder]

Ok, lets say I have a Slippery Bogle in play and then cast Rayami, First of the Fallen. I guess I don't know if replacement abilities use the stack or if there is a response window if I move to sacrifice the Bogle where my opponents can still target and remove Rayami. It feels like there isn't a window to target Rayami but.... I don't really know.

[lyonhaert]

Well, neither of Rayami, First of the Fallen's abilities are triggers. You said "then cast" Rayami, but since hexproof matters to the example I assume Rayami resolved and an opponent wants to hit her with spot removal. My understanding is that Rayami's first ability exiles the Slippery Bogle when you sacrifice it (assuming the sacrifice is really fast such as being a cost like in your example with Phyrexian Tower) with a blood counter on it. Then Rayami's second ability gives her hexproof as long as the bogle is exiled with a blood counter on it. They could respond again by removing the bogle from exile (such as if they use Pull from Eternity or could flash in Riftsweeper) to keep their removal from fizzling, but otherwise your plan would work.

[Argus]

Ok, lets say I have a Slippery Bogle in play and then cast Rayami, First of the Fallen. I guess I don't know if replacement abilities [sic] use the stack or if there is a response window if I move to sacrifice the Bogle where my opponents can still target and remove Rayami. It feels like there isn't a window to target Rayami but.... I don't really know.

Effects do not use the stack. This includes replacement effects.

405.6. Some things that happen during the game don't use the stack.
405.6a Effects don't go on the stack; they're the result of spells and abilities resolving. Effects may
create delayed triggered abilities, however, and these may go on the stack when they trigger (see
rule 603.7).
405.6b Static abilities continuously generate effects and don't go on the stack. (See rule 604,
"Handling Static Abilities.") This includes characteristic-defining abilities such as "[This
object] is red" (see rule 604.3).

Also, there is no "move to sacrifice". You just sacrifice.
The sequence would look something like this:

1. Your main phase begins.
2. You get priority and cast Rayami, First of the Fallen. You pay the cost using 1BGU obtained in an unspecified way.
3. You get priority and pass.
4. Opponent gets priority and passes. (If they had something like Cancel, they could cast it at your spell.)
5. The top object on the stack (the spell cast in step 2) resolves. You put Rayami, First of the Fallen onto the battlefield under your control.
6. You get priority and activate the "T, Sacrifice a creature: Add BB." ability of Phyrexian Tower. You pay the cost by tapping Phyrexian Tower and sacrificing Slippery Bogle. Slippery Bogle would go to your graveyard, but instead you exile it with a blood counter. Rayami now has hexproof. You add BB to your mana pool.
7. You get priority and pass. (Or maybe you have plans for that BB, but you didn't describe them.)
8. Opponent gets priority. Opponent might want to cast something that targets Rayami, but that has hexproof.