Can you plot pure function without specifying variable?

Yes, you can plot it, but not using Plot. For example, you could map the function over a range of values and then use ListLinePlot:

With[{xmin = 0, xmax = 4π},
 ListLinePlot[f/@Subdivide[##,100],DataRange->{##}]&[xmin,xmax]
]

This uses the new function Subdivide with 100 plot points.

The reason why Plot requires you to specify a dummy variable is that it takes expressions and not functions as its argument. Therefore, the plot variable is not identifiable by a slot, and you need to specify it by naming the plot variable.

There is the function Unique, which is often used to avoid symbol name collisions.

You could use it like this:

Plot[f[#], {#, 0, 10}] &[Unique[]]

Of course, in another sense, it's the most arbitrary that a variable can get (it's just a variable with a name of the form $12345 where 12345 represents some arbitrary number).

The short answer is no, not with Plot, which like many other Mathematica built-in functions that range over a set of values, is designed to follow a standard syntax pattern for which Table may be considered the prototype. Consider Do or even Manipulate$\text{*}$, which was added long after Plot. Both follow this pattern although in the case of Manipulate the semantics are wildly different.

$\text{*}$ See this conference article for a discussion of how Manipulate was deliberately designed to emulate Table. It also has many additional interesting things to say about Manipulate.