Ask Mr. Smarty Plants

Is this a test?  Slightly to our surprise we actually did find some plants that live up to your specified requirements. Some were not altogether reasonable, such as Magnolia grandiflora (southern magnolia). Evergreen, with glossy green leaves and big, white, fragrant flowers, it is unfortunately in the 72 to 100 foot class, and would probably not be suitable for a pot. So, scratch that.

We'll list the plants we found in two categories, and then give you the one that comes closest to filling the bill. We are recommending only plants native to Central Texas (with one exception, and it's close), because the Lady Bird Johnson Wildflower Center promotes the care, protection and propagation of plants native not only to North America but the area in which they are being grown. Because native plants are already adapted to an area's rainfall, temperatures and soils, they will require less fertilizer, maintenance and water. These plants were all selected for full sun, which we regard as more than 6 hours of sunlight daily. You can follow the plant links to the individual webpage on each plant and find out its projected height, water needs, etc.

NICE FLOWERS BUT...

Ageratina havanensis (Havana snakeroot) - pink to white flowers, attracts butterflies and hummingbirds, blooms October and November, but deciduous.

Anisacanthus quadrifidus var. wrightii (Wright's desert honeysuckle) - Also called Flame Acanthus, blooms red, orange June to October, loved by hummingbirds, but deciduous.

Pavonia lasiopetala (Texas swampmallow) - blooms pink April to November, but deciduous.

Hibiscus coccineus (scarlet rosemallow) - Texas Star Hibiscus, but not really native to Texas. Native to Louisiana and other southern states, so that's close. Bright red blooms July to September, but dies back to ground in the Fall, will regrow from roots in Spring.

Echinacea purpurea (eastern purple coneflower) - pink purple flowers, blloms April to September, but dies back to rosette in Fall.

EVERGREEN BUT...

Chrysactinia mexicana (damianita) - Evergreen but with lots of small flowers, aromatic foliage, blooms yellow April to September.

Leucophyllum frutescens (Texas barometer bush) - Evergreen (or ever gray-green), beautiful pinky purple small blooms intermittently throughout the year, depending on rainfall.

Coreopsis lanceolata (lanceleaf tickseed) - Evergreen, blooms with daisy-like yellow flowers April to June.

Tetraneuris scaposa var. scaposa (stemmy four-nerve daisy) - evergreen, lots of small yellow flowers virtually all year.

Wedelia texana (hairy wedelia) - evergreen in southern part of range, blooms orange, yellow May to November.

NICE FLOWERS, EVERGREEN IF...

Hibiscus martianus (heartleaf rosemallow) - large red flowers January to December if there is no frost. 

Magnolia grandiflora

Ageratina havanensis

Anisacanthus quadrifidus var. wrightii

Pavonia lasiopetala

Hibiscus coccineus

Echinacea purpurea

Chrysactinia mexicana

Leucophyllum frutescens

Coreopsis lanceolata

Tetraneuris scaposa var. scaposa

Wedelia texana

Hibiscus martianus