Yep, the fruiting bodies of some sort of fungus, aka mushrooms.

Bird's Nest Fungus, one of the artillery fungi.

Bird's Nest fungus doesn't "shoot" the way the Artillery fungi do, and they're not closely related.

The Artillery fungi are in the genus Sphaerobolus, while Bird's Nest fungus is Cyathus olla.

Bird's Nest fungus here:
http://waynesword.palomar.edu/pljune96.htm

A bit about both here:
http://www.personal.psu.edu/faculty/d/d/ddd2/

My first thought (before I put my reading glasses on) was that someone was playing a prank and put reese's cups in their container!
But I see now that I shall be disappointed, for it's not chocolate peanut buttery goodness, only a fungus!

shrooms are one of natures decomposers ... and unless literally attached to the plant.. are usually of no significance but knowledge ...

it indicates that your potting media has some type of unfinished compost in it ... which isnt bad itself... usually it is wood of some kind ...

the only down side is.. that as the stuff in your media break down.. it will change the properties of your media ... that which was once well draining media.. MIGHT become less so ... if left for years ... which you should do with potting media anyway .... its one of the reason we repot potted plants every so often ./..

the fact that your does this with a lot of water... MIGHT mean ... you are on the verge of retaining too much water ... some plants mind that.. like trees ... which prefer a much higher drained media ...

i mix pro mix type media with 50% wood chips ... for very high drainage ... because that is what the type of plants i grow in pots want ... they are usually much drier than yours ... but in the rainy season... i do get the shrooms.. of various sorts.. as noted above ...

ken