Chris,
Since this was not a dream then I can't really say what it was that caused this circled number to appear. Like you I also have the numbers 666 in my ss number interposed by one other number. And like you I do not subscribe to biblical interpretation of the number 666 unless the person is very devote to their religion {and then only as a contributing factor and not the prime meaning}. The best I can do is provide analysis the what you saw after awakening from a dream/nightmare, perhaps that having something to do with the number's appearance. I will amplify the images and actions as if it were a dream to see what message might be.

Dream Title: Woke up with #6 on my arm
-woke up-awaken to an unknown qaulity/aspect about self
-#6-equilibrium/balance/harmony in physical life
-on my arm-an extension of yourself associated with nurturing

-number #6 on bicep with circle around it-conflict in developing equilibrium/ balance/harmony/wholeness/not fully developed
-not indented but skin was raised-security/protection not elevated in physical life

As for the number 6, the symbolic representation is
From my dream dictionary at Myths-Dreams-Symbols
(1) May be number for completeness. Six is a multiple of three (same as three).
(2) May symbolize conflict.
From my Power of Dreams Dictionary
-cooperation, balance, tranquility, perfection, warmth, union, marriage, family, and love. Your mental, emotional, and spiritual states are in harmony. It is also indicative of domestic bliss.

As for the circle it can represent symbolizes perfection, completeness, immortality and/or wholeness. In Jungian psyche it represents wholeness, completion. Reaching the part of yourself that has integrated all aspects of your being; physical, psychological, spiritual.

Does the above amplification fit somewhere in your life? Was 'nurturing' an issue in childhood {parental}? Is there an aspect of/in your life where completion or wholeness is an issue or goal? That is how I would look at the images and what they might mean if they were in a dream.

Another note on the number 666. Manly P. Hall states it this way as it has to do with biblical applications {one I can agree with}.
"All higher numbers can be reduced to one of the original ten numerals, and the 10 itself to 1. Therefore, all groups of numbers resulting from the translation of names of deities into their numerical equivalents have a basis in one of the first ten numbers. By this system, in which the digits are added together, 666 becomes 6+6+6 or 18, and this, in turn, becomes 1+8 or 9. According to Revelation, 144,000 are to be saved. This number becomes 1+4+4+0+0+0, which equals 9, thus proving that both the Beast of Babylon and the number of the saved refer to man himself, whose symbol is the number 9. This system can be used successfully with both Greek and Hebrew letter values."

The number 6 is a reversed number 9. Number 9 is the number for the universe {actually it is 432,000 with the numbers added together equaling 9}. 666=18. My position is the number 6 represents wholeness of the physical self/life and 9 the spiritual {which would include the physical}

Jerry

Age & Gender & Location {Required}: 67 Male Cocoa, Fl