Self or Ourselves or Oneself

The proper word for this is ourself, which is the singular reflexive pronoun corresponding to we, because each of us is a singular construction, even though us is first person plural.

This word is in the Lexico dictionary (although it is not used very often):

ourself: first person plural

1. Used instead of “ourselves,” typically when “we” refers to people in general rather than a definite group of people.
   [reflexive] ‘we must choose which aspects of ourself to express to the world’

You could use ourselves in Lexico's example, but only ourself works well in the OP's sentence.

Short of including the somewhat clunky himself / herself form, the only really "correct" version here is the "gender-neutral"...

Each one of us thinks about themself¹ all the time.

Even native speakers get slightly confused in contexts like OP's example here (see The hidden flaw in “singular they”—what to do about reflexive pronouns?, as asked previously on ELU). So you will often encounter the plural reflexive pronoun coupled with the singular verb form thinks about themselves¹ in such contexts. But imho that's not "correct".

¹ Those are both links to many written instances in Google Books, but I must acknowledge that according to this NGram, plural themselves is actually far more common overall.

Each one of us thinks about oneself all the time. - correct but old-fashioned

Each one of us thinks about himself/herself all the time. - correct but clunky

Each one of us thinks about ourselves all the time. - incorrect, mixed singular / plural

Each one of us thinks about self all the time. - correct but technical

or rephrase:

We all think about ourselves all of the time. - how I would write it.