Crossed & nested random effects model Here comes my case:
I conducted an experiment with roughly the following design:
30 participants, each with a unique id, were asked to rank using a likert scale how much they liked images of forests. All the participants ranked the first 8 images and then the following 5 images were randomly drawn from a pool of 15 images. Therefore, in total each participant viewed 13 images of forests, but not all images were viewed by each participant. As the responses are ordinal I have gone with a cumulative link mixed effects model to preserve the structure of the data.
And here comes when I need your expertise. So far, I believe that the random terms of my model should take into account that:
Participants (id) Image (id)
However, Im confused about how to incorporate the random effects as each participants views some but not all of the same images. Thus-far I have come to the conclusion and coded in R package ordinal it as:
clmm(likert_Rating ~ Experience + X.4 + X.3 + (1 | part_id) + (1 | Plot_ID), 
     data = TotalF)

However, no matter how many models I try, I am never sure about how to include the randoms effects using this design.
I would really appreciate if some of you could point me in the right direction as I am struggling to decide how to include the random effects.
 A: Indeed, it seems that you have a crossed design, expecting that ratings from the same subjects will be correlated and that ratings for the same image will be correlated. Hence, the model you have specified seems logical. Note that the model does not require that all participants rate all images (i.e., that you have a complete balanced design). It will also work with an unbalanced design.
Check also this section of the GLMM FAQ for advice on how to code the part_id and Plot_ID variables.
A: I think that what you are doing is fine. Abstracting the situation at hand, we do not have fully crossed random effects. Nevertheless we can specify the model the same as if it were a fully crossed design. Ultimately a "deal-breaker" would be a random effect having an inadequate number of levels (usually <5) which is not the case here. One might want to check  Schielzeth & Forstmeier (2008) "Conclusions beyond support: overconfident estimates in mixed models" and their subsequent 2010 paper "Cryptic multiple hypotheses testing in linear models: overestimated effect sizes and the winner's curse" for some issues that might arise by overconfident estimates and potentially misspecification of the random effects.
