This very matter is  discussed in  depth by Mathias,  in Chapter 9 of his  **The Stength of Mac Lane  Set Theory** https://www.dpmms.cam.ac.uk/~ardm/maclane.pdf. There he shows  that to prove 

  $\;\;$   "for all $n$ there exist $n$ pairwise nonequinumerous infinite sets" 

requires some use  of  unbounded Separation, and to prove 

  $\;\;$   "there exists an infinite set of pairwise nonequinumerous infinite sets" 

requires some use  of Replacement. He also discusses various refinements in connection with formula complexity and stratifiability. For example, Coret showed that that the stratified instances of Replacement are already theorems of Zermelo set theory, whence "requires some Replacement" entails "requires some unstratifiable Replacement". 

Algebraists might prefer these assertions concerning sequences of  $\mathbb R$-linear spaces defined though duality: $L_1=\mathbb{R}[t]$ and $L_{k+1}=L_k^*$. In this setting, 

   $\;\;$  "for all $n$ the sequence $L_1,\ldots, L_n$ exists"  

 requires some use of unbounded Separation,  and 

  $\;\;$   "the sequence  $L_1, L_2, \ldots $ exists"

 requires some use of Replacement.



Whether or not these assertions count as ordinary mathematics, I find them considerably easier to grasp than Borel Determinacy, which seems vastly more intricate. Then again, maybe the second example counts  as "there's life beyond $V_{\omega+\omega}$". 


Meanwhile, some of the motivation of this question resonates with mine in  posing these questions:

http://mathoverflow.net/questions/120598/when-must-it-be-sets-rather-than-proper-classes-or-vice-versa-outside-of-fou

http://mathoverflow.net/questions/117910/can-one-exhibit-an-explicit-kuratowski-infinite-set-without-invoking-replacement

Some of the  comments  on the  first (Sets vs Classes) allude  to various constructions in homotopy theory  involving long-running transfinite recursions - and even  large cardinals - so presumably there is some Replacement involved.