Analysts had thought the $1.35 level would be a line in the sand for the European Central Bank's tolerance of euro strength, but recent comments from officials have suggested the central bank is unlikely to intervene in the near-term.
On Friday, Dutch Finance Minister Gerrit Zalm said the euro's rise was still within acceptable margins, suggesting little official alarm at the pace of the single currency's gains.
The euro was a touch firmer at 140.40 yen after rising on Monday to 140.74 yen, its highest level since June 2003 and close to an all-time high.
Analysts are now waiting to see what reaction policymakers have to the euro's latest surge. Some analysts reckon the ECB may be reluctant to pick a fight with the markets unless currency moves become disorderly.
"ECB officials are hoping to set up the euro as the number two currency after the dollar, so they are likely to be cautious (about intervening)," said Hideaki Inoue, chief manager of the forex division at Mitsubishi Trust and Banking.
Some analysts generally believe Japan has become more tolerant of a stronger yen and would prefer to intervene in coordination with other countries.
Japanese data on Tuesday showed industrial output rebounded less than expected in November while deflation deepened. However retail sales beat forecasts in November while the unemployment rate fell to 4.5 percent -- the lowest in six years.
So far in 2004, the dollar has shed nearly 8 percent versus the euro and the British pound and nearly 4 percent against the yen.