I can state the first two rows in a flash although I sometimes stumble on Neon. After that it’s an educated guess.
Thanks to Tom Lehrer, I can recite about half the names (the first verse). Plus the last line of the second verse.
They are not, of course, in order as they appear on the table.
The first row, and that’s it.
In high school I had an odd fascination with the PTE and even made my own. I think at that time I could’ve recited several elements and perhaps even a few atomic weights and electron shell numbers.
Um, Adamantium? Carbonadium? Duranium? Mithril? Tritanium? Vibranium? Do I get extra points for listing in alphabetical order?
Two minutes ago, when I read this and tried, I got to manganese. So a third of the way into the fourth row, but it’s been a while since last I worked on memorizing it.
I could usually make it to the start of the lanthanoids when I recited it more regularly.
A few years back, I decided to memorize the periodic table. I didn’t recite it but rather typed it into a file. I got so I could consistently do all but a small handful of elements and a few times did it without error. That only lasted a couple weeks and now I can’t do any better than before. So it didn’t take. No doubt age is the problem. If I were younger, I’m sure more would stick. (The same thing happened when I tried to memorize the Mexican states some time before that.)
My 11th grade Chemistry teacher gave us a mnemonic for the two lines after H-HE
The first is:
LiBeBCNOFNe (pronounced LieBeeBuhKuhNoffNee)
The second is:
NaMgAlSiPSClAr (pronounced NahMigAlSipSklar)
They just roll off the tongue, don’t they?
When it comes to reciting the periodic table, I’m just not in my element.
I made a rhyme that does not rhyme or make sense, but it still works for me after decades: Lirio Bello Besa Conmigo No Oses Falaz! Neon goes without saying. Then put H (and deuterium and tritium, if you are feeling bold) and He on top, left and right. From there I could do the first two columns and the nobel gases. I can’t anymore. 
Will try again soon.
I can remember being able to recite all the elements back in the early 70’s. Does that count? The PTE has gotten longer and I’ve gotten older so reciting stuff like that just doesn’t happen anymore.
Only to Nitrogen, because Meg recites it that far in A Wrinkle in Time.
I know a lot of the ones near the bottom end in -ium.
If I’m trying to just list them in order, not very far (when I just tried it, only to sodium (11)). But I could do much further if I were filling in a blank table, even if I had to first draw the blank table myself: Possibly as far as rubidium (37) before the first gap, and a fair number past that as well.
I’d be pretty hopeless on 104 and above, as those weren’t yet discovered at the time I was staring at periodic tables in high school. But those elements don’t belong on the periodic table, anyway, since the periodic table is about chemical properties, and those elements are too sparse and short-lived to even have chemical properties.
35 years ago I could recite the whole shebang, minus some trans-uraniums that have been synthesized since, of course. Now I stall out somewhere around Vanadium just into the 4th row.
Hydrogen. I’m done.
This, except I probably wouldn’t get to Rubidium. I might be able to get a few dozen boxes right, though since the right and left hand sides are easier to remember beyond the first few boxes. I’d probably either forget which rows Cu, Ag, Au, and Pt are in or would remember correctly and get all 4.
True Story
A few years ago, I was in the Philly March For Science. IIRC It was in defense to likely actions and policies that would happen after Trump became President. At one point, somebody yelled
“Let’s chant the periodic table!”
I yelled “H!”
Somebody else yelled “H! E!”
The original chanter yelled “I was just kidding and was unprepared for this!”
It was a good day.
I got to sodium and thought calcium came next (it is magnesium) but was stuck after that.
No, because you missed Unobtanium 
.
Dammit