A penthouse apartment should be at least two of the following things: expensive, and possessed of great views and the backdrop for events that are out of the ordinary.

Recently a few top-floor flats with terraces — a terrace being the key penthouse feature — have been doing their best to fulfil all these requirements.

Take, for example, the $7 million Upper East Side penthouse bought by Bernie Madoff, the swindler, from his Ponzi scheme profits. He is now residing at the more spartan Manhattan Correctional Center.

Another glamorous penthouse in North Kensington housed the contestants for The Apprentice and has become, for some of the show’s millions of fans, an object of envy.

Each morning the contestants set out from this canalside complex to carry out Sir Alan Sugar’s commands. They return at night to recover from a boardroom encounter with the testy tycoon or an acerbic aside from his sidekicks, Nick Hewer and Margaret Mountford.

Filming of The Apprentice having finished, this Portobello Dock development of 12 apartments is now for sale through a number of estate agents, including Harrods Estates, at a price of £10.5 million, reduced from £12 million. You can, however, experience life in the complex by becoming a tenant (according to Gumtree.com, rents range from £650 to £850 a week).

The Battersea apartment belonging to Brian Balfour, an art dealer, fulfils the first two penthouse criteria. The vast 3,600 sq ft property, in the shimmering Albion Riverside building between Battersea Bridge and Albert Bridge, is on the market for £4.75 million through Charles McDowell Properties — making it suitably expensive.

The four-bedroom home also has the requisite unencumbered views from its front terrace (there is another at the back) over the Thames. The highlights of this panorama include historic Cheyne Walk in Chelsea, the Victoria and Albert Museum’s tower, One Hyde Park, the Knightsbridge development where one penthouse fetched £100 million — and Wembley Stadium.

Gazing at the river and rhapsodising over sunsets is a pleasant way to pass the time, but other riverside developments in London also provide ample opportunity for revery.

Sadly, few of these blocks are themselves worth prolonged scrutiny because of their often bland 1990s styling. Albion Riverside, a silver-clad, crescent-shaped building, is a visual treat inside and out, however, having been designed by Foster and Partners, Lord Foster of Thames Bank’s firm.

The lobby of the building and the flats are the clean, uncluttered spaces that you would expect from this architectural practice.

But for Balfour, who buys and sells the works of Damien Hirst and Lucian Freud among others, the Foster style was not sparse enough. When the father of two bought the penthouse two years ago, he demolished walls, removed the recessed lighting and stripped down surfaces, revealing the steel and concrete.

The peeling away of the layers of paint on the metal staircase that leads up to the sitting room took one workman more than a week. The lack of reverence towards Foster’s oeuvre seems to be a typical trait; Balfour says he specialises in the big-name artists such as Hirst partly because the marketing has already been done for him.

The result of the deconstruction is a live-work space of the deluxe species: Balfour closed his gallery before the economy turned down and has run his business from home. Thanks to the stock of artworks displayed around the apartment, it is not a generic early-Noughties minimalist pad.

Ironic rockeries constructed from stones that Balfour picked up at low tide from the riverbank have been placed on both terraces. No wonder that his teenage children think it all delightful. Balfour is selling to travel more, particularly in the Far East. It will be up to the well-heeled owner of the penthouse to decide whether just to enjoy the view, or put the address on the map.

An equally colossal (3,745 sq ft) London penthouse, used as a showcase for the owner’s own art collection, is for sale via Knight Frank for £6,95 million (the works, which include a Damien Hirst, are not included). The apartment, in the Halcyon building in Holland Park, was created by knocking together two adjoining flats: the owner bought the first in June 2008 and the second two months later.

Knight Frank, 020-7938 4311 Charles McDowell Properties, 020-7581 8357