Before the days of blinkenlights and fancy vacuum tubes, humans still found creative ways to compute.
This list is in the range fingers/mental < ancientcomputing < eniac, so fingers, pure brainpower, and charts aren't included, nor are any electronic computers past tabulators and the bombe. i.e. no
NewMath Computers.
The dictionary definition of computing (i.e.
http://dictionary.reference.com/search?q=computing) has "To determine an amount or number.", which would include measuring devices, so maybe our definition of computing is limited by assuming algorithms. "Computing" might very well include sundials and thermometers. Simple devices/computers have their own category.
Human scale
 Abacus  For arithmetic
 Addiator  For arithmetic
 Arithmometers  For arithmetic
 Astrolabe  Astronomical computer
 CharlesBabbage's DifferenceEngine and AnalyticalEngine  For arithmetic
 Egyptian water clock  For time
 Clypsedra  For time
 The AntikytheraDevice
 Liebniz's calculator  For arithmetic
 Notturnale  Astronomical
 Pascal's box  For arithmetic
 Salamis Tablet  For arithmetic
 SlideRule  For arithmetic/logarithms
 Verge and foliot  Clock mechanisms
 CurtHerzstark?'s CurtaCalculator Pocket Calculator (1938)
 planimeter  2D area from 1D perimeter
Large scale
 Analematic sundial  For calculating time based on the month and location of sun
 FoucaultsPendulum  Rotation of earth
 Avebury and Stonehenge, Ring, Rollwright Stones, Long Meg, Mayburgh Henge,France's Carnac, Stones of the Hebrides
ElectroMechanical
 Hollerith Tabulator  For tabulating census data (on HollerithPunchCards)
 Enigma  For encrypting german world war 2 information
 Turing's Bombe  For breaking enigma encryption codes
 Would KonradZuse's machines fit in here?
 VannevarBush's DifferentialAnalyzer?
Static devices/"Simple" computers/Grey area
 Sundial  For time
 Weathercock  Weathervane
 Merkhet  For time
http://amo.net/AtomicTimeZone/help/TimeHistory.html

 "The Egyptians improved upon the sundial with a merkhet, the oldest known astronomical tool. It was developed around 600 B.C. and uses a string with a weight on the end to accurately measure a straight vertical line (much like a carpenter uses a plumb bob today). A pair of merkhets were used to establish a NorthSouth line by lining them up with the Pole Star. This allowed for the measurement of nighttime hours as it measured when certain stars crossed a marked meridian on the sundial."
But note that, due to precession, the pole star is not fixed:

 "For the ancient Egyptians, the Pole Star was not Polaris, but Thuban in Draco, while observers in the far future will see yet other stars at the Northern Celestial Pole"
http://www.glyphweb.com/esky/concepts/northerncelestialpole.html
2030: When I was young, programming was actually done in this country. I even saw a programmer once.
 When I was young, there was actually more than one country.
 When I was young, there was actually more than one party.
 When I was young, we only held parties on the weekends.
See also Digital Computer Museum
http://edthelen.org/comphist/1981Catalog.html