Nemu
When Mayuri created his 'daughter', he spent a fair amount of time considering her dæmon's form. Since the shinigami aspects of her genetics were modeled after his own (with adjustments to keep her compliant, of course), he originally hypothesized that it would be an aye-aye. However, as the experiment progressed and he made more changes, altering her DNA until it barely resembled that of a shinigami, he began to wonder if she would even have a dæmon. After all, dæmons only accompany human souls and he was doing his best to make sure she would embody only the most useful human traits – everything else could be discarded, including the nuisance of having a dæmon.
In fact, by the time of Nemu's awakening, he was hoping his modifications would keep a dæmon from forming. Had, in fact, done extra experiments to force that result.
At first, he thought the procedures were successful – he saw no sign of a dæmon during Nemu's first three days. His 'daughter' was submissive, unemotional, scientific…everything he'd hoped for. Other shinigami were uncomfortable in her presence and some decried her creation as unethical, but they were uneducated cretins, unworthy of even pondering his genius.
On the fourth day, he was ordered to deal with Zaraki (the barbaric imbecile needed a new eye patch) and he placed Nemu in charge of keeping Yachiru out of trouble.
"Hi! I'm Yachiru!"
Nemu blinked placidly at the small girl. "I am Nemu."
"Let's play!"
"What is 'play'?"
"It's where you have fun, silly! Let's play tag!"
"What is 'tag'?"
"Like this!" She whacked her dæmon on the back of his head and took off. "Tag! You're it!"
"No fair!" Vesper was instantly at her heels, crashing through everything in his way.
"Vice-captain, please desist." No response. "Vice-captain-."
"Tag..oops!" Vesper lunged at Yachiru from a high bookshelf…and tripped. He found his fall arrested by a pair of gentle, furry arms and blinked up into the big orange eyes of an unfamiliar lemur dæmon.
"Who're you?"
The stranger merely set him down and vanished into the shadows. His black/grey/white coloration allowed him to blend seamlessly with the patterns of light and darkness.
Two curious pairs of eyes looked to Nemu for answers.
"His name is Benjamin." No one, not even her father had met him yet. Benjamin did not like being around other people; inside the lab, he remained on top of high equipment/bookshelves and outside he kept to the rooftops, stopping in eaves and under overhangs. He was quick and clever with his paws, but shy and quiet.
"Time to go, kids." Kenpachi stumped back into the room.
"Aw, but we wanted to meet Benny!"
"Who th' fuck is Benny?"
Mayuri overheard the explanation. In response, he threw a screaming tantrum and locked his daughter inside, threatening intercision and a host of painful dæmon-related experiments.
However, Nemu was more useful to him alive and intact, so in the end he was forced to leave her, and Benjamin, alone.
Mayuri's Dæmon
Species: Aye-aye (Daubentonia madagascariensis)
Name: Nyx (Night/Daughter of Chaos)
Nemu's Dæmon
Species: Ring-tailed Lemur (Lemur catta)
Name: Benjamin (Son of my Right Hand)
Yachiru's Dæmon
Species: Unsettled
Name: Vesper (Evening Star)