I cannot think of a mechanism that would entirely prevent a gene from being regulated. For example, consider mechanisms like histone modification: there is very little about the sequence of a single underlying gene that can itself cause or prevent histone modification, yet those changes regulate the expression of associated genes. However, you can really only provide evidence in science for things that happen; providing evidence that things do not happen is often questionable. If you have some example gene and you'd like to say "this gene is not regulated", the best you can ever get to is "I haven't yet found a circumstance in which this gene is regulated by any manipulation I know of".
In practice, there are some genes that are not typically "switched on/off" and always expressed at fairly constant rates, we call these housekeeping genes. For many of these, the consequence of 'switching them off' would be death of the cell. However, I would not consider that these genes are "not capable" of being regulated, rather, I would say that they are specifically regulated to be always active, and that there is very strong evolutionary pressure for this to occur. To show that I mean by this, consider some quoted lines from Wikipedia:
The housekeeping gene expression levels are fine-tuned to meet the metabolic requirements in various tissues. Biochemical studies on transcription initiation of the housekeeping gene promoters have been difficult, partly due to the less-characterized promoter motifs and transcription initiation process.
Little is known about how the dispersed transcription initiation of housekeeping gene is established. There are transcription factors that are specifically enriched on and regulate housekeeping gene promoters. Furthermore, housekeeping promoters are regulated by housekeeping enhancers but not developmentally regulated enhancers.
In summary, steady activity is carefully controlled, and difficult to study. Comparatively, if a gene has very different expression in different environments, you can follow an iterative process to look at cells in each environment and see what is different: if you find another protein is phosphorylated or otherwise modified, or has also changed expression, you might be looking at a transcription factor involved in your gene of interest. On the other hand, if something never changes, where do you start? Trickier problem for an experimentalist.