Festuca - Ornamental Grasses - The dwarfish Festuca Glauca is the common blue version, but there are several Festucas that are worth considering - especially for the additional Autumn effect.
Fritillaria - Imperialis - Meleagris - At the bottom of this list, simply because it is an alphabetical list! Bulbous perennials with two basic growth habits - being the upright showy Crown Imperial Lilies, and the more subtle but delightful Snake's Head Fritillaries. Both types - and others are welcome in most garden situations.
Farfugium - The first time I encountered a Farfugium was a cold day in November when I was wandering for some late Autumn colour photos. I saw it from a distance - and not easy to miss with its striking evergreen varieagated foliage. It was beside a pond, and it is best planted in damp areas or as a marginal aquatic perennial.
Ferula - Giant Fennel - (NOT the Herb)Giant because it can reach a height of several meters - 2m normally the minimim! The green one is spectacular, but there is also a less vigorous bronze variety. Well worth growing , as the foliage is light, so in spite of height, can be absorbed into most gardens.
Filipendula - Similar to, and often mistaken as - Astilbes. But, Filipendula are quite distinct. Several upright clump forming varieties with feathery foliage held above attractive foliage, but also a few low growing mounds.
Foeniculum - Fennel Herb - Very attractive foliage on this vegetable/herb. Well suited to the perennial border until time to harvest it!
Fragaria - Ornamental Strawberry - Interesting members of the strawberry family grown for attractive flowers and low growth habits. Some edible fruits are an added bonus.
Francoa - Attractive evergreen perennials clump forming habits, and shades of pink for flowers. NOT the hardiest, but ok in sheltered placed, or if given a bit of protection for the winter.
Fuchsias - Botanically they are shrubs, but as most of the hardy ones are better cut to ground level in the Spring, they are worth a mention for addition to the herbaceous border. There are some perennial types, but almost impossible to source.