Web Graphics

Texture Town - So many textures
Smolpxl - Lots of little pixel graphics
Geoblinkies - Search for Blinkies and Gifs
Bank of Pixels - Many Pixel Graphics
The 88x31 GIF Collection - Button Gifs
Web Badges World - Bunch of web badges
Blinkies Maker - Make your own blinkies
Gifs Paradise - Gifs sorted by type
Animated Images - More gifs
Gif Gallery - Even more gifs
99 GIF Shop - Even more more gifs
A.N. Lucas's Button Collection - Buttons from 1990's and 2000's
lapin's oldweb gfx hoard - A bunch of old graphics from the 90s/00s
snailspng - Cut-out pngs
Artwork Graphics - Just a ton of different graphics
Pixel Safari - Many pixel graphics
Radiotrophic - So many blinkies
GifCity - A carrd page with a ton of different web graphics
Decome Tumblr Tag - Tons of web graphics on tumblr
KaomojiKuma - Kaomoji gifs
Cake Batter - Lots of favicons, and some dividers and backgrounds
Adrian's Blinkie Collection - Lots of blinkies, stamps and buttons

Helpful for Websites

Bit Font Maker - You can make pixel fonts with this
Hotglue.me - I haven't used this, but in theory it seems useful
flamingtext.com - Make cool text for your website
Glow Txt - Another cool text generator
EZ Gif - You can use this for resizing and editing gifs
Pigment by ShapeFactory - Interesting for making colour palettes
88x31 Web Button Maker - Make your own web buttons
Pixel Art Scaler - For upscaling pixel art so it doesn't look blurry at a bigger scale
Pet Pet Generator - Make a petting gif
Online Image Editor - Edit images/gifs online
DitherIt - For dithering images
RV's free Javascript/DHTML scripts - A bunch of popular free-to-use scripts
Add an interactive game character on your webpage - In theory you can program an entire game into your website
Transparent Textures - So many textures
Font Collector - Free fonts
Collection of Pixel Backgrounds - A tumblr page with tons of pixel backgrounds
Label Maker - Make cool labels
Background Archive - Tons of tileable backgrounds
vinizinho's free backgrounds - Cool original tileable backgrounds
eggnored on tumblr - An insane amount of tileable backgrounds

Fun Stuff

windows93 - Old windows computer with tons of stuff to do
neal.fun - Games and fun stuff
Pixel Planet Generator - Generate little planets that you can save as pngs or gifs
Pixel Space Background Generator - By the same person, you can make tileable space backgrounds
Pokefarm Q - Pokemon egg hatching game
PokeClicker - Pokemon clicker game
HORG.com - A faux-scientific archive of bread clips

Other

Tyoma's Cool Links - This is a massive link list with so many helpful things
FreakPhone Web Resources - Another link list with a bunch of helpful things for website making (the graphics section is my favourite)
SlashDiv Links - Another link list with lots of links to check out
Foollovers - It's in Japanese but its not too hard to navigate, lots of pixel graphics and website templates
Pokemon Sprite Index - Tons of pokemon sprites, both images and gifs
Spriters Resource - Many many many sprites from many games