Ah, one of my google results was for the RHS website where, within the description of Phyllody, they mention other related conditions. It seems that this little plant has "flower proliferation" :
"What is proliferation?
Proliferation is a disorder that usually affects the flowers of a range of plants including clovers, the daisy family, opium poppies, plantains and roses. Bellis perennis ‘Prolifera’ – the hen and chickens daisy – is a stable cultivar that exhibits the phenomenon reliably with flowers arising from within the main flower. Occassionally a triple-tiered effect with flowering arising in the second set is seen.
"The opium poppy ‘Hens and Chickens’ is an example of where secondary flowers are formed around the main flower.
"Another type of proliferation is occasionally seen in plants with solitary flowers. Certain roses seem particularly prone, with the flower stalk appearing to grow through the flower to produce another flower, creating a two-tiered effect .
"Symptoms
The symptoms of proliferation become apparent when;
"The flowers continue to producte new growths beyond the stage at which growth usually ceases
Within the flower multiple buds appear around OR within the open blooms
Not all the blooms on the plant are usully affected; in repeat-flowering roses for example, the second flush is usually free from problems
Control
Prune off the affected flowers and the next flower produced by that stem should be normal. However, there is no necessity to remove affected shoots if they are considered pleasing.
"Cause
The cause of proliferation is unknown but, when it appears unexpectedly, it is probably due to a random genetic mutation at some point during the development of the flower.
"It is possible that physical injury may be involved, perhaps by a late frost.
"Where proliferation occurs each year, a virus might be the cause and affected plants will have to be replaced."
) and they replied saying it's a not uncommon genetic flaw.
