Los Blancos clinched the double last term after winning their 10th Champions League crown over city rivals Atletico Madrid and also lifted the Copa del Rey at Barcelona's expense, with Bale scoring a stunning winning goal.
Atletico claimed the domestic title, however, and Bale admits his side are desperate to win every trophy available in the coming campaign - beginning with the European Supercup on the Wales star's home soil next month.
"We enter six competitions and we want to win every one," Bale said.
"That would be amazing. We are hungrier than ever.
"No team has retained the Champions League so we want to do that.
"And there is a special touch for me because the Super Cup (against Sevilla) is in my home town (Cardiff). Hopefully we can do the job and get the trophy."
Real have signalled their intent to win more titles next season with the signings of World Cup-winner Toni Kroos and Colombia star James Rodriguez this summer.
The latter was unveiled as a new Blancos player this week after completing his move from Monaco in a deal reported to be worth €80 million.
With Uruguay star Luis Suarez having also joined Barcelona from Liverpool earlier this summer, Bale believes that the influx of some of football's best players to Spain shows the strength of the top flight in the country.
"The Spanish league is the most exciting and the world's top players play there," he said.
"Barcelona or Real Madrid are always there, near the Champions League final or winning it.
"The Premier League is a great league but we are attracting the best players to La Liga.
"It is great to be involved in it."
Real kick off their pre-season when they play Inter in the International Champions Cup tournament in California on July 27.