An object is gunky provided that all of its parts have proper parts. Gunk is usually considered a really outré possibility. I want to offer some examples of intuitively conceivable gunky objects to broaden the philosophical imagination. The examples are all predicated on an Aristotelian ontology that allows for parts but denies other aspects of classical mereology. The thought behind the Aristotelian ontology of parts is that the parts of a thing correspond to natural functionally delineated subsystems. My heart is a part of me, as is my left arm. But there is no such part of me as "the left half of my heart" or "me minus my left arm".
Example 1: An infinite tree in 3D.
Here's a plausible Aristotelian thought about trees. Suppose that we have a branch A that splits into sub-branches B and C. Then branch A is an object that has both B and C as parts. However, there is no such part as A minus (B plus C). I.e., there is no object that consists of the part of A before the split. For the naturally delineated subsystem is the whole branch, including sub-branches, rather than the part of the branch without the sub-branches. Now imagine a fractal tree-like structure where the branches split into sub-branches, and the sub-branches into sub-sub-branches, and so on ad infinitum. Suppose, further, that there are no smaller natural functionally delineated subsystems than branches, sub-branches, sub-sub-sub-branches, etc. (This differs from real-world trees, which are made of cells.) The result is gunky: each part of the structure is a branch at some level, and each branch itself gives rise to sub-branches.
Dynamically, the structure can be thought of as built out of extended simples. We start with a trunk (a zero-level branch) that grows gradually. Then the trunk splits into branches. As a result, the trunk ceases to be simple: it has two or more proper simple parts, namely the branches, but it is not just the sum of the branches. The branches initially are simples, but eventually split themselves. If each step takes half the time of the preceding, after a finite amount of time we have the full infinite gunky tree.
Example 2: A four-dimensional example.
Suppose a spatial simple can survive becoming non-simple.(This was a governing assumption in the dynamical story in Example 1.) Suppose there are no proper temporal parts. Now, imagine we have a simple A, which survives becoming a non-simple made of two simples B and C. Then repeat the process with each simple. Continue ad infinitum, but don't require the process to speed up in any way. At any finite time, there are only finitely many objects. But the whole four-dimensional thing is gunky: A is made of B and C, B is made of D and E, and so on.
Example 3: Aristotelian temporal parts of a spatially simple thing.
On the Aristotelian ontology of parts, there won't be arbitrary temporal parts: there won't be the temporal part of me from my third to my fourth year. However, there might be naturally delineated temporal parts, like my adult part. Now imagine a person who never dies, and every five years receives a PhD in another discipline. If PhD-in-discipline-X counts as a naturally delineated temporal part, then the person will have a sequence of temporal parts like: doctor of biology, doctor of physics, doctor of chemistry, etc. Moreover, if we list these parts in the correct order, it gets gunk-like. If her first PhD is in biology and the second is in physics and the third is in chemistry, then the doctor of chemistry will be a part of the doctor of physics which will be a part of the doctor of biology. Moreover, there might be no such part as not-a-doctor-of-biology or not-a-doctor-of-physics (by the same token as on the Aristotelian story, there is no such part as me-minus-my-left-arm). Now, suppose that the person in question is an angel and hence has no spatial parts, and that the person has no significant temporal divisions besides the acquiring of PhDs. Then the individual is gunky: each part has a proper part. And this is easy to imagine, as long as we aren't worried about temporally extended simples.
Final remark: I don't know if these conceivable things are metaphysically possible.