What is the highest viewing spot in London that is accessible free of charge?

The Walkie-Talkie (20 Fenchurch St) is only the 13th-tallest building in London at 160m height, but it does have a free viewing gallery (height 155m):

The famous enlarged glass dome of 20 Fenchurch Street is dedicated to three storeys of exquisitely landscaped public gardens and London’s most exclusive social spaces, including observation decks and an open air terrace.

http://skygarden.london/sky-garden

Although it's free, you do have to book your tickets online in advance.

Image © User:Colin, CC BY-SA 4.0

Be warned that the 'sky garden' has not found universal favour, so don't set your expectations too high:

The building was crowned with a Sky Garden, a babylonian jungle in the clouds that would be the pride of the Square Mile, framed as not just a place for bankers to drink, but a public space accessible to all. The reality is anything but. If you book three days in advance, or reserve a table at one of the overpriced dining concepts, you can go through airport-style security and be treated to a meagre pair of rockeries, in a space designed with all the finesse of a departure lounge. A hefty cage of steelwork wraps around in all directions, obscuring much of the view, while the restaurants rise up in a boxy stack of glass portable cabins. The more you pay, the worse the view gets: at the very top of the gourmet ziggurat, you’re as far from the windows as possible.

Carbuncle Cup: Walkie Talkie wins prize for worst building of the year, The Guardian, 2 Sept 2015

The fourth-tallest building in London (225m) is the Cheesegrater (The Leadenhall Building). It's not normally open to the public but is opened up for one weekend a year during Open House London.

Open House was started in 1992 as a small, not-for-profit organisation to promote public awareness and appreciation of the capital's building design and architecture. The intention was to open up London's splendid buildings to the general public who don't otherwise have access. We saw this as a way of helping the wider community to become more knowledgeable, engage in dialogue and make informed judgements on architecture.

The dates for Open House London in 2016 are 17 & 18 September

You'll need advance tickets - check the Open House website nearer the time (in past years info has become available in August).

Image © User:Colin, CC BY-SA 4.0

I believe the Gherkin (30 St Mary Axe, 180m) also opens its doors for Open House weekend.

If you don't mind being outside, you can get some good views over most of London from some of the hills that surround it. Two good spots are Parliament Hill, in Hampstead, and Alexandra Palace, in Muswell Hill.