If we change the question to "what is the minimum number of elements common to every form of life?", my lists would be as follows. (I am not considering either viruses or prions to be 'alive').
The following elements are surely common to all forms of life (List 1):
H (1), C (6), N (7), 0 (8), Mg (12), P (15), S (16) [#7].
In addition, the following elements are very likely to be common to all forms of life (List 2):
Na (11), Cl (17), K (19), Ca (20), Fe (26), Cu (29), Zn (30) [#7].
If we broaden things to include elements occurring in any form of life I would add (List 3):
B (5), F (9), Si (14) [!Seemingly], V (23), Cr (24), Mn (25), Co (27), Ni (28), Se (34), Mo (42), Cd (48), Sn (50) [!Maybe], I (53) , W (74) [#14].
No element in either list has an atomic number greater than 74 (Tungsten). (In the list given by Frieden (1972), the highest is I (53)).
For List One (and as pointed out by @bob1), H (1), C (6), N (7), 0 (8), S (16) are all present in proteins. P (15) is present in ATP and NAD. We must surely also include Mg (12), as the true substrate of almost all ATP-requiring enzymes is a complex of Mg++ and ATP. I can't see that any element can be removed from this list, but it could be argued that we need to add some from List 2.
For List 2, Fe (26) is present as haem in haemoglobin and the cytochromes, and (not necessarily complexed to heme) in other components of the electron transport chain. It might be possible that anaerobic life is possible without iron, in which case Fe (26) needs to be demoted to List 3 (but I doubt it). Zn (30) occurs in many enzymes (including yeast and liver alcohol dehydrogenase). It might be possible for a life form to exist without requiring Zn, but I don't know of any examples.
If we restrict consideration to plants, then Mn (25) needs to be promoted to List 1 (Mn is an essential component of Photosystem II, the enzymic system responsible for the 'splitting' of water in photosynthesis). As life as we know it would be impossible without PS, Mn (25) should possibly be in List 1 without qualification.
List 3 will probably always be contentious, but here are my thoughts:
This, I think, justifies B being included in List 3: required for some forms of life (but not a minimum requirement for life).
F (9) occurs in fluroacetate (See also The natural production of chlorinated compounds by G W Gribble)
Si (14). A paper in Science in 1972 suggests that Silicon is an essential element for the chick (news to me!). Frieden (quoted below) has this to say: "Possible structural unit of diatoms; recently shown to be essential in chicks. "
V (23). Occurs in haloperoxidases (See Coordination environment changes of the vanadium in vanadium-dependent haloperoxidase enzymes. The vanadate ion is very similar to phosphate. See The role of vanadium in biology)
Cr (24) Whether Chromium is an essential trace element is controversial. It may play a role in insulin action (see Wikipedia) but this is not universally accepted (the Wikipedia article gives a good summary of the current position). Some bacterial can reduce chromium but this seems to be a detoxification mechanism (See also Chromium: is it essential, pharmacologically relevant, or toxic?). Frieden states "Essential in higher animals; related to action of insulin." but this, it appears, is still controversial. Possibly Cr (24) needs to be deleted from List 3.
Co (27) is a component of Vitamin B12
Nickel (28). Urease, the first enzyme crystallized (by Sumner in 1926), an accomplishment that played an essential role in establishing that enzymes are proteins, was shown in 1984 to contain Nickel (see Urease (Wikipedia))
Se (34) Selenium (as selenocysteine) is directly incorporated on the ribosome into some enzymes (SelenoCys is considered "the twenty-first amino acid")
Mo (42) is a component of quite a number of enzymes, including xanthine oxidase and is often associated with a pteridine cofactor. See Molybdenum and Molybdenum-Containing Enzymes
Cd (48) A carbonic anhydrase from the marine diatom Thalassiosira weissflogii contains cadmium. It is thought that in oceans depleted of Zn, diatoms can use Cd as a zinc substitute. (See Biochemistry: a cadmium enzyme from a marine diatom and
Structure and metal exchange in the cadmium carbonic anhydrase of marine diatoms)
Sn (50) There have been some reports that Tin is an essential trace element for rats and humans (see Case study: bioavailability of tin and tin compounds and Frieden (quoted below)), but the evidence appears to be very scanty. Tin probably should be deleted from List 3.
I (53). Iodine is known to be a component of the thyroid hormones
W (74) Tungsten The first enzyme to require tungsten, a NADP-linked formate dehydrogenase from Clostridium thermoaceticum was purified and characterized in 1983. Since then a number of tungstoenzymes (at least ten) have been identified, including acetylene hydratase. In many cases (and in a similar fashion to molybdoenzymes), W is often bound to a pterin cofactor. (See Tungsten in biological systems)
I think The chemical elements of life by E.Frieden, published in Scientific American in 1972, is still relevant, and I have relied heavily on the table on p55 of this publication in compiling the above lists
I haven't considered Sr (38), Nb (41), Ba (56) or Ta (73) but from Tungsten in biological systems it appears that there is a case for these elements to be included in List 3