Here is a list of current paying and validated faucets, which allow you to start with a given cryptocurrency. This list is based on a research which took me approx. 5hrs to present you actual working and tested faucets, tell me in the comments if you found additional ones which you can recommend.
Bitcoin
| URL | Description | Payouts |
|---|---|---|
| Moon Bitcoin | Old and trusted faucet. | Weekly direct payout (25k Satoshi) |
| wbit.co | Old and trusted faucet. | Direct payout from 6,000 Satoshi, 250 Satoshi per hour. |
| jetcoin | Faucet from 2015. | Up to 750 Satoshi per claim, payout from 35,000 Satoshi. |
DOGE (Dogecoin)
| URL | Description | Payouts |
|---|---|---|
| Moon Dogecoin | High reputation faucet. | Weekly direct payout (if over 100 DOGE) |
LTC (Litecoin)
| URL | Description | Payouts |
|---|---|---|
| Moon Litecoin | High reputation faucet. | Weekly direct payout (if over 500,000 litoshi) |
I kept the list short on purpose, to only present you with actual working faucets. There is a lot of scams and bad sites out there, which also often present you with malware, so this compiled list should be comprised of actual working and paying legit faucets. Also let me know in the comments if you find anything fishy.
A good hint for getting a multitude of coins with an online wallet and faucets for most of them: try POSwallet. POSwallet is a website which has around 99 coins (including Bitcoin) and offers you wallets, so you can get started with coins such as NAVcoin, BlackCoin, etc. - sadly "regular" (non proof-of-stake coins) are not supported there (except Bitcoin), but it is a good place to start nevertheless.