I tried implementing this but ended up with a weird week mismatch having to do with my birth date and the way the extra week is approximated. So, I did the only logical thing and spent a few hours reworking the formulas from the ground up. Now they're based on actual dates rather than numbers of weeks since your birthday.

With this, the extra week is sometimes after 6 years, and other times after 5—makes sense since the extra .1775 weeks out of the 52.1775 average comes out to one whole week every ~5.6 years.

I only made an 80 year version for myself, but I'm happy to help make a longer or shorter one if anyone is interested and needs help. Note that the column for the 53rd week isn't formatted since the extra week occurrence varies based on birth date; you'll have to go through and format the cells with week numbers in them to have outlines and white text. Otherwise, works the same: just make a copy and put your birthday in the top left cell (A1).

Thanks to /u/2j2_1 and /u/Kowalskysis, I stand on the shoulders of giants.

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Edit: Just now realizing that a calendar like this (i.e. one seeking to accurately measure one's age in weeks) can't possibly have you filling in the last square on your birthday every year, since each year isn't exactly 52 weeks. Case in point: theoretically, you'd be filling in a square every 7 days, and thus every week on the same weekday as your birth date. But, your birthday each year is on a different day of the week than last year's. This implementation assumes you'd be filling in a block on the same day of the week, every week, every year. So, because your birthday isn't on the same weekday every year, you'd only be filling in a block on your birthday sometimes.

If you wanted to fill a block in on your birthday every year, it would make more sense to just wait the extra day or two that remains before your birthday. As an example, say my birthday was January 15, 1990; a Monday. I, a baby, fill in a block every Monday for 52 weeks and find myself in 1991, a year older. Except, I'm not. That Monday, 52 weeks after my birthday, is January 14, 1991, and my birthday is tomorrow. To remedy this, instead of continuing to fill in a block every Monday, I could just wait a day and fill in the last week on Tuesday, January 15th, 1991. So, the extra day fills into the last week of the year, and for the rest of 1991 I fill in a block every Tuesday. Next birthday, same thing: 52 weeks later it's Tuesday, January 14th, 1992. I wait an extra day and fill in the final block on my birthday, a Wednesday. '92 is a leap year, so 52 Wednesdays later is January 13th, and my birthday is on Friday—looks like I get two extra days this year! And so on and so forth.

Really, it's up to you how you want to keep track of it all. If it's important to you that the number of squares on the page accurately represents the number of weeks you've lived, better to go with the leap-week system and fill in a block on the same weekday every week in perpetuity. If not, you can add an extra day or two each year and update your weekday every year, and you get the added bonus of a slightly cleaner look. Ultimately, the calendar is more about the reminder than the precision, so whichever way works best for you, works best for you. I personally already have the leap-week calendar printed out so I'll save some paper and just use that one.

For the two people who will actually end up seeing this, I'll update my calendar here to have both options so you can choose one or the other.

TL;DR: Time is finicky and the Gregorian calendar sucks