Can you name the most common antigen that cancer cells in general can't live without?
This is a hard question to answer because each type of cancer is a whole different game at the molecular level, however there is something you might be interested in called "hallmarks of cancer" (image 1) which are key features for cancer to thrive. They were originally described by Hanahan and Weinberg (2000) and have been revisited by Fouad and Aanei (2017) in an updated review. The hallmarks of cancer aren't specific genes themselves, but rather broader categories that englobe certain types of genes.
If you want to look at specific genes, we could talk a bit about what more purely data driven approaches have found. There is a very interesting work by Kandoth and collaborators (2013) that presents data from The Cancer Genome Atlas (TCGA), which is a database of different kinds of genomic features from several different cancer types. Among the results of this work (specifically on figure 2 which I include here as Image 2) you can see a table of 127 significantly mutated genes across 12 different cancer types. One gene that is well known in medical literature for being frequently mutated in almost any cancer type (situation which is also observed in the aformentioned table) is TP53 which encodes a protein that is key for genome maintenance in normal cells by participating in DNA damage and repair (which is a cancer hallmark under the "altered stress response" category). In the table you can also spot tissue-specific cancer genes such as APC in colon/rectum cancer and VHL in renal cell carcinoma.
Just to make sure you can understand the table, the gene names are presented in the right (rows) and the names of the cancer types are presented at the top (columns). The description for the cancer types abbreviations can be found here. To the left of the table there is a functional classification of the genes so you can know more or less what a given gene does in a quick manner. The values presented are simple frequencies (divide the number of samples of a particular cancer type that have the mutation over the total number of samples of that same cancer type) expressed as percentages (so they are already multiplied by 100). The final column (Pan-Cancer) shows the same idea, but taking all samples from all cancer types into consideration for the frequency calculation.
So for answering your question regarding if there is a common feature in all types of cancers: strictly speaking at the gene level no, since even the TP53 mutation which is the most common amongst all cancers might not be present in some cancers; however, looking at it by categories of genes (as one or some of the 7 hallmarks) reveals that all cancers apply similar strategies to thrive (they exhibit different mutated genes, but these genes frequently belong to the same biological pathways and participate in similar functions). I hope you find this useful (:
Image 2: frequency of mutation by gene and cancer type.
Fouad, Y. A., & Aanei, C. (2017). Revisiting the hallmarks of cancer. American journal of cancer research, 7(5), 1016–1036. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5446472/
Kandoth, C., McLellan, M. D., Vandin, F., Ye, K., Niu, B., Lu, C., … Ding, L. (2013). Mutational landscape and significance across 12 major cancer types. Nature, 502(7471), 333–339. doi:10.1038/nature12634
National Cancer Institute. TCGA Study Abbreviations. Retrieved from https://gdc.cancer.gov/resources-tcga-users/tcga-code-tables/tcga-study-abbreviations