Here's my experience.

First, I'll admit I have not tried this on that particular kind of foam. I did this with the 2" pink foam and foam board from Dollar Tree and PECO code 83 track. I used DAP Alex caulk. All surfaces were painted. The caulk is spread very thin, just enough to show. No oozing allowed. And for a suggestion. Paint the surfaces before gluing. From what I did, the paint adheres to the foam. And the caulk adheres to the paint. Because looking at your pictures, none of your attempts was successful - because the glues attacked the foam. For permanence, I did try Loctite PL 300. That won't come up and is a total success for me. The DAP Alex is not permanent. But for track laying/so I could pry it up when necessary, the Alex was great. For you, the Loctite. And paint first. I firmly believe that would be a good way to test. Finally, I'm just trying to help. In no way am an expert. In fact, when I tried working with SPS, I got so much mess, I had to give up. Those little bits of foam are a pain to get up. Vacuum or brush. And they got everywhere.

Foam adhesive?

   You can save time by just going to the store and buying a caulking gun type tube of foam safe adhesive. It's a common construction product. BTW,  white glue and yellow glue don't dry well when sandwiched between foam sheets as they need air to dry. ......DaveB

To clarify the point of the test ...

Morgan Bilbo, and DaveB,   Thank you both for replying, but I think ya'll missed the exact point of this test, beginning with the third sentence of my original post.  

I'm not trying to solve any problems with joining foam to foam; I've got that covered with construction adhesive. Rather, I'm looking for an adhesive, for adhesive sake (that does not eat foam), but more specifically an adhesive that can be used as a filler for holes / voids in the foam. That's all explained above.

Morgan,  Not sure what you're seeing in the photos, but only the E6000 ate the foam. Everything else worked very well for the first test. The caulk you see in my test is Alex Flex.  It would actually work for what I'm trying to achieve, as a filler.  But for this I need a very small, fine-tipped applicator to get into my voids. I don't think I can get there with a 9" caulk tube applicator.

Back to my second photo, and the real point of my experiments: I've figured out what's happening with the fluid level dropping with the wood glue and clear gorilla glue. When I drilled those holes for testing, I just grabbed a large drill bit and made holes to be filled.  That big 'ole drill bit really made a mess of the inside of the holes - created lots of big voids by ripping out foam beads.  In other words, the drill bit created lots of little nooks and crannies inside the holes for the glues to fill.  Hence, the fluid levels dropped.  None of them are actually eating the foam.  That Tacky Glue is very thick straight from the bottle and isn't as "fluid" as the clear Gorilla glue and the wood glue.  I'm seeing the level drop very slightly on the Tacky Glue but nothing like the other two.  

I'm testing the Tacky Glue again in a different hole drilled using a Dremel cutting bit.  The edges of the hole are very slick and doesn't offer any places for the glue to "escape" the hole.

Will advise ...

Caulk

OK. I did miss understand. However, DAP Alex is what I use to adhere roadbed and track. I'm also able to pry it up for repair. If I want permanent, I go to Loctite PL 300. And for what I think you are trying to do either should be sufficient. If only because it won't hurt plastic. Now, as for tubes. Can't you just transfer the caulk from the tube into something with a smaller tip that will reach where you want? And you didn't comment on my suggestion of painting the SPS. ?? It might help the SPS to firm up where you need it to hold whatever it is. ? I'm just not clear I guess. Can you give a picture of the actual location? I'm missing why you need holes in the first place. Your experiments don't show any holes - other than the ones you apparently accidentally got. DaveB was apparently just telling you what I am trying to convey. The Alex is cheap and has a huge number of uses for me. I just like working with it. It's not as easy as applying plastic cement from a needle applicator, but it does work for me in a lot more places than I had anticipated.

Filling holes

  Something like sculptamold or fixall works fine.Plain old sheet rock taping compound also....DaveB

I'll try one more time

Morgan Bilbo commented,

I don't want to apply any sort of paint to the SPS.  

My experiment did indeed show holes that I created - here they are again.  The photo below shows three holes that I drilled into the foam.  The holes are 1/2" round and 1/2" deep.  The holes are filled with Tacky Glue, wood glue, and Gorilla Clear Glue.  The point of this experiment was two-fold:  1) To determine which of these adhesives worked best as a filler - literally to fill holes that I will bore into the foam, which I will explain again in a moment.  And, 2) to determine if any of these adhesives would eat the foam if used to fill holes.  None of these adhesives ate the foam.  The tacky glue performed best as a filler.  

Now, to explain the purpose of the holes again: My structure will be sitting on a bunch of brick piers. I will bore holes into the foam for the brick piers to sit in.  I will then use an adhesive to fill around the brick piers.  Image if you dug a round hole in the ground, placed a 4x4 post in the hole, and poured concrete around the post to keep it secured in the hole ... that's what I'm doing with the adhesive ... using it like concrete around my brick piers.

I have a way to go before I need to do this.  I will likely experiment with various other "fillers" before I make a final decision.  Those experiments will include caulk, only if I can figure out a way to get the caulk into some sort of "micro applicator". 

Glue or filler?

 They are different products designed for different uses. Sounds like your easiest solution would be to drill the holes then put some caulk or plaster in them then push the poles in and clean off any excess filler that squeezes out. Don't over think it. You could spend days looking for methods or a few minutes just doing it with what you have on hand....DaveB

If you are filling holes...

... you might try foam. 

Great Stuff Expanding Foam that is. It will completely fill voids and then you can carve the excess (there will be excess) down to the shape you need. 

https://www.greatstuff.dupont.com/products/greatstuff-gaps-cracks.html

That foam in the pic looks like the stuff made of

little beads.  If you don't paint it, I think your are going to have little beads breaking loose and appearing all over the railroad.  If you don't want color or any shine, at least paint it with something like dull coat or an acrylic flat finish to seal it.

Painting the foam ... kind of

It occurred to me that I will actually be painting this foam, kind of, near the end of the process.  Well, at list tinting the foam before as I apply the turf.  Woodland Scenics sales a product called Scenic Cement for applying turf, and I think that stuff can be tinted.  I've tested with diluted PVA for applying the turf and that worked very well (and sealed the foam in the process).  Just need to test the same PVA/water mix tinted with acrylic paint.

That's what I said!

LOL I mentioned paint mainly because of those "beads" that will make a mess. I do admit to being confused about what you are doing. But my experience with foam is that I paint it. Is it necessary? Not always, but for me, it's easier in the long run to go ahead and paint stuff. You can always re-paint it another color if you need to.

I'm not sure I understand the

I'm not sure I understand the issue... Any (latex) water based, water cleanup, material should be foam safe.  If the instructions say to clean up with solvent then it will more than likely eat the foam.  Silicone may be the exception.

Most caulk and glue will shrink when filling large (1/4" plus) size holes. If shrinking is an issue for the intended application one should use a plaster or non shrink filler material.

Glue isn't for filling holes,

Glue isn't for filling holes, it's for being spread in thin layers between close fitting pieces and attaching them to each other.

To fill in holes in foam, use lightweight spackling.... Filling holes is what it's meant for, it's water-based, and its exactly the same stuff that Woodland Scenics sells as "foam putty", for exactly that purpose.

Because I hope to use something "liquid"

Glad for all the input.  I'm not rejecting any ideas for fillers - I'm just hoping to find something in liquid form that'll harden well.  

Here's a look at the bricks I'll use to build the piers under the house. I'll have ~ 15 piers that'll be ~ 10mm wide x 10mm wide.  I'm guessing that I'll create ~13mm holes to set the piers into.  I'm looking for something in liquid form that I can "pour" around the piers with a fine applicator tip.  To me, that's sounds like the easiest way to fill the gaps around the piers - just pour something into the gaps and walk away.  I'm hoping that I can avoid using anything that requires blades/scrapers to apply (putties, spackle, etc.) so I can avoid getting that material all over my bricks.  

Soooooo, I tested with what I had on hand that was liquid and would harden.  Those liquids just happened to be glues/adhesives.  The only problem I had with any of those glues/adhesives were they sort of seeped into the foam.  The Tacky Glue did the best, being so thick in consistency and all.

Here’s an idea you may want to think about.......

........13 mm is just a hair over 1/2 inch.  Drill 1/2 inch holes in your foam where you want your piers and put 1/2 inch wood dowel material into them gluing them into place.  Then build your piers on top of the dowels.  You can recess them a bit from the top so ground cover will hide them. 
 

The reason I suggest this is I built a station canopy (see below picture) which was then inserted into holes I drilled into the really rigid blue foam used as the base of the platform covered with thin styrene for concrete, and they wobbled all over the place. The foam would not keep them solid.  I then had to drill out the holes a bit and glue plastic tubing in place.  Once I slid the posts into the plastic it was nice and solid with no wobble.

(This picture shows it in a wood jig)

Just a thought........

Peter

Appreciated

Appreciate your effort here, always good to add to the dataset / knowledge base. 

Rather than straight wall  smaller diameter holes to set your piers, have you considered either carving out shallow but larger diameter depressions or holes? I am thinking you may have better control by setting grade manually rather than trying to hold the piers at  an elevation in a fluid, no matter how viscous.

You could take a page from heavy civil construction here. If using the beaded board then you will not be able to get a flat bottom to your excavations and hold a consistent elevation, so you dig / drill a wee bit deeper than needed - then place a leveling product such as the tacky to your desired sub-grade elevation. Once cured then place your piers and when all check  out as level, "set" them by backfilling the surrounding shallow depression with the same product to lock 'em in place.

Hope I described that well.....

Bill

Last bit of advice.....

....on this.  Why are you using the beaded foam for a base when the rigid blue or pink foam is much stronger as a base material?

Peter