For example, when I have...

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...is it possible to remove the current page? I suppose it's doable by editing the "breadcrumb" template but I'm not sure what exactly to delete there. Any ideas? Thanks!

Add to custom.css:

.ipsSubNavigation li:not(.ipsMenu_item):not(.ipsMenu_sep):last-child,
.ipsSubNavigation li:nth-last-child(2) .fa.fa-angle-right {
    visibility: hidden;
}

Now the problem with this method is that it will always remove the current page's breadcrumb. So there will be no breadcrumb on the home page. That's what you want, however? If so, this is better than editing templates.

Add to custom.css:

.ipsSubNavigation li:not(.ipsMenu_item):not(.ipsMenu_sep):last-child,
.ipsSubNavigation li:nth-last-child(2) .fa.fa-angle-right {
    visibility: hidden;
}

Now the problem with this method is that it will always remove the current page's breadcrumb. So there will be no breadcrumb on the home page. That's what you want, however? If so, this is better than editing templates.

​

This exactly what I wanted, thank you so much. 

One thing though... this also removes the "More" dropdown menu with Activity Stream, Staff. etc. Any idea why?

Managed to fix that by editing your code to:

.ipsBreadcrumb > ul > li:last-child, li:nth-last-child(2) .fa.fa-angle-right {
    visibility:hidden;
}

Thnkas once again!